Generalizing higher-order Bessel-Gauss beams: analytical description and demonstration

Damian N. Schimpf; Jan Schulte; William P. Putnam; Franz X. Kärtner

doi:10.1364/OE.20.026852

1. Introduction

Bessel-Gauss beams offer properties that Gaussian beams do not possess. For instance, they exhibit an extended depth of field, and away from a strong on-axis intensity region they exhibit an annular intensity distribution in the far field [1, 2]. These attributes have made these beams interesting for applications in biomedical imaging, material processing, particle trapping, or laser-based acceleration of particles [3–7].

Laser resonators supporting Bessel-Gauss beams have also been designed and implemented [8–10]. More recently, enhancement cavities based on Bessel-Gauss-type modes have been proposed for strong-field applications, such as cavity-enhanced high-harmonic generation (HHG). Bessel-Gauss modes offer the benefits of near-perfect out-coupling of the generated intra-cavity high-harmonics and an increased intensity ratio from the focus to the cavity mirror surfaces [11]. In these applications the well-known Bessel-Gauss beam has been exclusively considered as the operating mode. Having observed higher-order azimuthally symmetric mode solutions with numerical mode-solvers, the most pertinent questions we ask are: Do higher-order Bessel-Gauss beams exist? How can they be described analytically and demonstrated experimentally?

The existence of higher-order Bessel-Gauss beams showing azimuthal phase dependence (i.e. carry optical angular momentum) is known [12, 13]. However, azimuthally symmetric higher-order Bessel-Gauss beams are hitherto not reported in the literature, to the best of our knowledge.

In the present paper, we first analytically describe higher-order Bessel-Gauss beams. Based on these findings, we then design an optical resonator and show numerically that it supports modes resembling azimuthally symmetric higher-order Bessel-Gauss beams. Finally, we demonstrate these higher-order Bessel-Gauss modes experimentally.

2. Generalizing higher-order Bessel-Gauss beams

Conventional Bessel-Gauss beams can be built up by a superposition of decentered Gaussian beams, whose centers are positioned on a circle and whose beam directions are pointing to the apex of a cone [14, 15]. In the following we will show that higher-order Bessel-Gauss beams can be formed in a similar way by using the more general decentered Hermite-Gaussian beams as component beams for the superposition on a single cone.

First, we define decentered Hermite-Gaussian beams and describe their propagation through optical systems. After that we superpose these component beams to the higher-order Bessel-Gauss beams and also discuss their propagation properties. Finally, we show that these beams can collapse to the azimuthally symmetric Laguerre-Gaussian beams.

2.1. Decentered Hermite-Gaussian beams

Decentered Hermite-Gaussian beams are a generalization of the conventional Hermite-Gaussian beam solution. The additional features are a displacement of their beam center by a vector r_d0 = (x_d0, y_d0) in the x–y plane and a tilt of the mean beam direction with respect to the z direction by an angle ε₀. They are also a solution to the paraxial wave-equation, and we explicitly define the decentered Hermite-Gaussian beam at the input of an optical system (z=0) as follows

\begin{array}{l} h_{m n} (x, y, z = 0) = & \frac{1}{w_{0}} H_{m} (\frac{\sqrt{2} (x - x_{d 0})}{w_{0}}) H_{n} (\frac{\sqrt{2} (y - y_{d 0})}{w_{0}}) \\ \times exp (\frac{i k}{2 q_{0}} [{(x - x_{d 0})}^{2} + {(y - y_{d 0})}^{2}]) exp (i k [ε_{x 0} x + ε_{y 0} y]), \end{array}

where q₀ stands for the initial Gaussian-beam parameter, w₀ is the Gaussian beam waist, which is related to q by

w = \sqrt{2 / k / Im [1 / q]}

. It is worth mentioning that in this contribution we express the complex fields with the exp(ikz) convention. We denote the transverse coordinate vector as r = (x,y). We also define a vector ε₀ = (ε_x₀, ε_y₀), which is the projection of k/|k| onto the x–y plane. The vector k/|k| stands for the mean direction of the beam, whose tilt with respect to the z direction amounts to the angle ε₀ = |ε₀|. The geometry is illustrated in Fig. 1.

Fig. 1 Geometry for the superposition of generalized decentered Hermite-Gaussian beams.

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It is noteworthy that the formulation of Eq. 1 is a special case of a previous definition of decentered Hermite-Gaussian beams [16].

2.2. Propagation of decentered Hermite-Gaussian beams through optical systems

The paraxial propagation of the decentered Hermite-Gaussian beam through an ABCD optical system is described by the Collins integral [17]. By exploiting the analogy to the solution for the conventional Hermite-Gaussian beam [17], we obtain for the final state of the propagation of the decentered Hermite-Gaussian beam:

\begin{array}{l} h_{m n} (x, y, L) = & \frac{1}{w} exp (i k L) exp (- i ϕ \cdot (m + n + 1)) \\ \times H_{m} (\frac{\sqrt{2} (x - x_{d})}{w}) H_{n} (\frac{\sqrt{2} (y - y_{d})}{w}) \\ \times exp (\frac{i k}{2 q} [{(x - x_{d})}^{2} + {(y - y_{d})}^{2}]) exp (i k [ε_{x} x + ε_{y} y]) exp (i φ) \end{array}

where L is the optical path along the optical axis, q is the new Gaussian beam parameter and w is the Gaussian waist. The vector r_d = (x_d, y_d) describes the new displacement of the Hermite-Gaussian component in the x–y plane at L, and the new tilt is described by the vector ε = (ε_x,ε_y). The parameter φ denotes a constant phase due to the transformation, the Gaussian Gouy phase is given by −ϕ · (m + n + 1). The new parameters are related to the initial ones via

(\begin{matrix} x_{d} \\ ε_{x} \end{matrix}) = (\begin{matrix} A & B \\ C & D \end{matrix}) (\begin{matrix} x_{d 0} \\ ε_{x 0} \end{matrix}); (\begin{matrix} y_{d} \\ ε_{y} \end{matrix}) = (\begin{matrix} A & B \\ C & D \end{matrix}) (\begin{matrix} y_{d 0} \\ ε_{y 0} \end{matrix});

φ = - \frac{k}{2} [C (x_{d 0} x_{d} + y_{d 0} y_{d}) + B (ε_{x 0} ε_{x} + ε_{y 0} ε_{y})];

q = \frac{A q_{0} + B}{C q_{0} + D};

ϕ = arg (A + B / q_{0}) .

Note that decentered Hermite-Gaussian beams transform in a very intuitive way: The geometric parameters, tilt and displacement, of the decentered Hermite-Gaussian beam follow the trajectories of meridional rays. The Gaussian beam parameters, q and ϕ, transform like that of an on-axis Hermite-Gaussian beam.

The decentered Hermite-Gaussian beams are formulated as a function of propagation parameters, i.e. the components of the ABCD matrix, in order to derive a propagation-dependent solution for the higher-order Bessel-Gauss beams in the next section.

2.3. Superposing decentered Hermite-Gaussian beams to higher-order Bessel-Gauss beams

We form the higher-order Bessel-Gauss beams by superimposing the component beams with the following attributes: the Hermite-Gaussian component is modulated only in the radial direction. The vectors r_d0 and ε₀ are collinear with r_d0= const. and ε₀= const. This corresponds to a superposition of beams whose mean centers are placed on a circle with radius r_d0 in the z=0 plane and whose mean beam directions point to the apex of a single cone with a semi-aperture angle ε₀, as shown in Fig. 1.

For the superposition, we express each component beam in an auxilliary coordinate system, which is rotated by an angle γ with regard to a fixed reference coordinate system (r,θ,z) so that its abscissa points in the direction of the vector r_d0. In this auxiliary system, the decentered Hermite-Gaussian beam consists solely of the H_m-term (i.e. H_n=0 = 1) and the abscissa coordinate is given by r·cos(θ−γ). The propagated decentered Hermite-Gaussian beam is expressed in the cylindrical coordinates (r,θ,z) as

\begin{array}{l} v_{γ} (r, θ, L) = & \frac{1}{w} exp (i k L) exp (- i ϕ (m + 1)) H_{m} (\frac{\sqrt{2} (r cos (θ - γ) - r_{d})}{w}) \\ \times exp (\frac{i k}{2 q} (r^{2} + r_{d}^{2} - 2 r r_{d} cos (θ - γ))) exp (i k ε r cos (θ - γ)) exp (i φ) . \end{array}

Because of the parallelism of r_d0 and ε₀, the following beam-propagation transformations hold

r_{d} = \sqrt{x_{d}^{2} + y_{d}^{2}} = A r_{d 0} + B ε_{0}

ε = \sqrt{ε_{x}^{2} + ε_{y}^{2}} = C r_{d 0} + D ε_{0}

φ = - \frac{k}{2} [C r_{d 0} r_{d} + B ε_{0} ε]

in addition to the relations for the Gaussian beam parameters, Eqs. (5) and (6). The formulation of higher-order Bessel-Gauss beams in terms of ABCD-matrix parameters allows us to study their transformation under the impact of optical components. We will expand upon this feature in section 3.1 describing optical resonators for these beam solutions.

The generalization of the Bessel-Gauss beam solution is found by integrating over the γ angle from 0 to 2π:

u_{m l} (r, θ, L) = \frac{2 π}{w} exp (i k L) exp (- i ϕ (m + 1)) exp (i φ) exp [\frac{i k}{2 q} (r^{2} + r_{d}^{2})] ℐ_{m l} .

The term ℐ_ml is given by

ℐ_{m l} = \frac{1}{2 π} \int_{0}^{2 π} d γ H_{m} (a cos (θ - γ) - b) \cdot exp (i α cos (θ - γ)) \cdot exp (i l γ),

where we substitute

α = k r (ε - (r_{d} / q)); a = \sqrt{2} \cdot r / w; b = \sqrt{2} \cdot r_{d} / w .

To generalize the analysis, we have introduced an azimuthal phase-variation while superposing the component beams. This phase is characterized by the index l. The integral ℐ_ml can be solved analytically (see Appendix), and the first four expressions are given by

ℐ_{0 l} = e^{i l θ} \cdot i^{l} \cdot J_{l} (α)

ℐ_{1 l} = \frac{e^{i l θ}}{2} \cdot i^{l} \cdot [2 i a \cdot ({(- 1)}^{l} J_{1 - l} (α) + J_{1 + l} (α)) - 4 b \cdot J_{l} (α)]

ℐ_{2 l} = \frac{e^{i l θ}}{2} \cdot i^{l} \cdot [- 2 a^{2} \cdot ({(- 1)}^{l} J_{2 - l} (α) + J_{2 + l} (α)) - i 8 a b ({(- 1)}^{l} J_{1 - l} (α) + J_{1 + l} (α))

+ (4 a^{2} + 8 b^{2} - 4) J_{l} (α)]

\begin{array}{l} ℐ_{3 l} & = \frac{e^{i l θ}}{2} \cdot i^{l} \cdot [- i 2 a^{3} \cdot ({(- 1)}^{l} J_{3 - l} (α) + J_{3 + l} (α)) + 12 a^{2} b \cdot ({(- 1)}^{l} J_{2 - l} (α) J_{2 + l} (α)) \\ + i (6 a^{3} + 24 a b^{2} - 12 a) \cdot ({(- 1)}^{l} J_{1 - l} (α) + J_{1 + l} (α)) + 2 \cdot (- 8 b^{3} - 12 a^{2} b + 12 b) J_{l} (α)] \end{array}

If m = 0 and l = 0, the well-known Bessel-Gauss beam is obtained [14]. Furthermore, the solutions comprising the terms ℐ_0l (i.e. m = 0) are identical to previously defined higher-order Bessel-Gauss beams exhibiting an azimuthal phase-variation [12, 13].

Here, we introduce higher-order Bessel-Gauss beams with a radial index m > 0. In Fig. 2 we illustrate the effect of the radial index by plotting the azimuthally symmetric (l = 0) higher-order Bessel-Gauss beams for different orders m in the near and far-field. The parameters are w₀ = 200μm, r_d0 = 0, ε = 0.3°, and λ = 1040nm. In the far-field (here, calculated at z= 4 · z_R, where $z_{R} = k \cdot w_{0}^{2} / 2$ ), all beams show an annular beam with variations in the radial direction such that the number of nodes corresponds to m. If l > 0 these solutions exhibit azimuthal phase-variations, as displayed in Fig. 3. To highlight the impact of the azimuthal index l on the solution, we consider the superposition (u_m,l + u_m,−l) resulting in a cosine-variation over the angular coordinate.

Fig. 2 Absolute value of the azimuthally symmetric Bessel-Gauss beam at z=0 for l=0 and (a): m=0, (b): m=1, (c): m=2, and in the far-field (z= 4 · z_R); (d): m=0, (e): m=1, (f): m=2.

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Fig. 3 Absolute value of the superposition of higher-order Bessel-Gauss beams in the far-field, complementary to Fig. 2 (at z= 4 · z_R) (a): (u_0,1 + u_0,−1); (b): (u_1,1 + u_1,−1); (c): (u_2,1 + u_2,−1); (d): (u_0,2 + u_0,−2); (e): (u_1,2 + u_1,−2); (f): (u_2,2 + u_2,−2).

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How does the Bessel-Gauss beam solution depend on the parameters r_d₀ and ε? - A classification is known for the fundamental Bessel-Gauss beam [12]: Generalized Bessel-Gauss beam, ordinary Bessel-Gauss beam and modified Bessel-Gauss beam denote the cases (r_d0 ≠ 0, ε₀ ≠ 0); (r_d0 = 0, ε₀ ≠ 0), and (r_d0 ≠ 0, ε₀ = 0), respectively. We extend this terminology to the higher-order Bessel-Gauss beams. For the example of a Bessel-Gauss beam with m=2, we plot the different cases in Fig. 4. On the left side, the radial profile is plotted over the propagation distance. On the right side, we display the transverse beam pattern at the position of minimum Gaussian waist (here at z=0). Figures 4(a) and 4(b) show the absolute value of the amplitude for a generalized higher-order (m=2) Bessel-Gauss beam for w₀ = 200μm, r_d0 = 0.8mm, ε = 0.3°, and λ = 1040nm. The vertical blue line highlights the (z=0) position. For this beam type the intersection with the symmetry axis is shifted from the position where the minimum Gaussian waist occurs. Figures 4(c) and 4(d) displays the ordinary beam type. The intersection coincidences with the position of minimum waist. Figures 4(e) and 4(f) picture the modified beam type. Furthermore, if both r_d0 and ε₀ are zero, a azimuthally symmetric Laguerre-Gaussian beam is recovered, as displayed in Fig. 4(g) and 4(h).

Fig. 4 Beam propagation and transverse pattern (at z=0) for: (a) and (b) generalized higher-order (m=2, l=0) Bessel-Gauss beam; (c) and (d) ordinary higher-order (m=2, l=0) Bessel-Gauss beam; (e) and (f) modified higher-order (m=2, l=0) Bessel-Gauss beam; (g) and (h) azimuthally symmetric Laguerre-Gaussian beam of order p=1, respectively.

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2.4. Collapse of the solution to Laguerre-Gaussian beams for ε = 0 and r_d0 = 0

To show the collapse of the azimuthally symmetric higher-order Bessel-Gauss beams to Laguerre-Gaussian beams, we write the integral in Eq. (12) for ε = 0 and r_d0 = 0. For uneven m the integral vanishes due to symmetry, and for even m (= 2 · p) we find:

\frac{1}{2 π} \int_{0}^{2 π} d γ H_{2 \cdot p} (\sqrt{2} \cdot (r / w) \cdot cos (θ - γ)) = {(- 1)}^{p} \frac{(2 p)!}{p!} L_{p} (2 r^{2} / w^{2}),

This integration is performed by substituting t = cos(θ−γ) and applying [18]:

\int_{- 1}^{1} d t {(1 - t^{2})}^{α - \frac{1}{2}} H_{2 p} (\sqrt{u} \cdot t) = \frac{{(- 1)}^{p} \sqrt{π} \cdot (2 p)! Γ (α + \frac{1}{2})}{Γ (p + α + 1)} L_{p}^{α} (u)

which holds for Im[α] > −1/2.

Equation (20) describes the remarkable property that the azimuthally symmetric Laguerre-Gaussian beams are a subset of the presented novel beam class. Thus, the presented solutions are a more general description for azimuthally symmetric beams.

3. Comparing the Bessel-Gauss beam solutions to modes of optical resonators

In the following, we would like to compare the analytical expressions for the fundamental and azimuthally symmetric higher-order Bessel-Gauss beams with numerical results. Specifically, we simulate a resonant optical cavity that supports a conventional generalized Bessel-Gauss beam for the fundamental mode, and should exhibit higher-order Bessel-Gauss beams as higher-order modes.

3.1. Design of the optical resonator

Figure 5 shows a schematic of the optical resonator, it consists of a spherical mirror with a radius of curvature R and an axicon mirror with a base angle ε.

Fig. 5 Schematic of the optical resonator.

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Ray matrices can be employed for the design of the optical resonator. One roundtrip is described by

(\begin{matrix} A & B \\ C & D \end{matrix}) = (\begin{matrix} 1 & L \\ 0 & 1 \end{matrix}) (\begin{matrix} 1 & 0 \\ - 2 / R & 1 \end{matrix}) (\begin{matrix} 1 & L \\ 0 & 1 \end{matrix}) .

Geometrical optics determines the radius of the displacement and tilt of Bessel-Gauss beams during propagation, as expressed by Eqs. (8) and (9). At the axicon mirror, these parameters are described by (r_d0, ε) and (r_d0, −ε) before and after one roundtrip, respectively. Thus, the resonator length is given by

L = R - r_{d 0} / ε .

Gaussian beam optics determines the Gaussian beam parameter q, which must repeat after one round trip. At the axicon mirror, one finds

q = - i \sqrt{L \cdot R - L^{2}} .

The resonator is stable for L < R. The minimum waist for the Gaussian component of the Bessel-Gauss beam occurs at the axicon mirror and can be determined from q = −iz_R.

To summarize, the resonator length and the waist w₀ of the Gaussian component of the Bessel-Gauss beam solution are fully governed by the radius of curvature of the spherical resonator mirror, the base angle of the axicon mirror, as well as the radius of the ring of the generalized Bessel-Gauss beam on the axicon, r_d0. For the following calculations, we chose a radius of curvature R = 250 mm, ε = 0.5° and r_d0= 1.5 mm, which result in L ≈ 78 mm and w₀ ≈ 196μm for a light wavelength of 1040 nm. These particular parameters were selected since they approximate the experimental configuration which, in turn, was determined by the availability of base angles for the axicon mirror and radii of curvatures for the spherical mirror. It results in a ring radius of the Bessel-Gauss beam that avoids the round tip on the axicon mirror and results in a not too large ring radius on the curved mirror. This situation is advantageous as it reduces effects of surface variations on the resonator mode.

The spherical mirror serves as the coupler to the resonator. For the coupling to the resonator with an excitation beam, it is important to know the beam parameters at the plane backside of this mirror. Geometrical optics of the meridional ray, characterizing the propagation of the Bessel-Gauss mode, gives a semi-aperture angle ε′ ≈ n_m · ε, where n_m is the refractive index of the mirror substrate, and a ring radius r′_d ≈ r_d + Δ · ε, where the ring-radius at the front surface is given by r_d = ε · R and Δ is the geometrical thickness of the mirror, which is given by Δ = Δ_out − s(r_out), where Δ_out is the mirror thickness at the outer radius r_out and the surface function is given by $s (r) = R - \sqrt{R^{2} - r^{2}}$ .

3.2. Numerical mode solver

Diffraction can be described in terms of a Hankel transform, which we numerically implement by employing a quasi-discrete Hankel transform (i.e. a Fourier-Bessel series expansion) [19–21]. In the following, we solve for the azimuthally symmetric modes.

To robustly obtain the numerical results for the higher-order modes, we have developed a fast two-stage mode solver. In the first stage, the modes are guessed by the principal component analysis (PCA) and in the second stage, a Fox-Li method refines these inputs to the actual mode solution. In particular, a generalized (fundamental) Bessel-Gauss beam is used as the start for the first stage. However, we intentionally use a larger waist w₀ and ring radius r_d0 than the predicted analytical solution for the fundamental mode would possess. This beam is propagated for 20 iterations. A matrix is formed by the roundtrip vectors. The PCA looks for the least correlated vectors, so that a linear combination of these so called principal components recovers the original data. Thus, the principal components are similar to the mode shapes, and serve as the inputs for the Fox-Li algorithm, which typically requires several hundreds of iterations. Here, we use only 500 iterations. The mode is then determined by looking for the coherent resonance of the sum of the iterated fields, for this the n-th iterated field is multiplied by the n-th power of a phasor in which the phase-shift is varied between 0 and 2·π [22]. To calculate the next higher-order mode, the second stage is restarted with a different higher-order principal component.

3.3. Comparison of the numerical mode-solution to the Bessel-Gauss beam solutions

To demonstrate the accuracy of the analytical expressions, the Bessel-Gauss beam solutions of different radial orders m are evaluated and compared to numerical results at the axicon mirror, and at two z-positions outside the cavity. The mode field is propagated in free space from the axicon mirror to the on-axis crossing point at a distance −(R − L) via an intermediate position −(0.9 · R − L). In Fig. 5, these positions are denoted as P1, P2 and P3. Figure 6 shows a comparison between numerical results and the analytical solutions. It can be seen that the azimuthally symmetric higher-order Bessel-Gauss modes represent an accurate analytical model for higher-order modes in resonators supporting the (m=0, l=0) Bessel-Gauss beam as the fundamental mode. Conversely, the numerical results validate the existence of azimuthally symmetric higher-order Bessel-Gauss beams, and constitute a reference for the hitherto unknown analytical beam solution.

Fig. 6 (a–c) Absolute value of the amplitude of the fundamental (m=0) Bessel-Gauss beam, (d–f) m=1 higher-order Bessel-Gauss mode, and (d–f) m=2 higher-order Bessel-Gauss mode at distance 0, −(0.9 · R − L), and −(R − L) from the axicon.

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Figure 7 shows the free-space propagation of Bessel-Gauss beams of different radial orders m. The propagation starts at the axicon mirror (z=0), which corresponds to the minimum waist point of the Gaussian component of the generalized Bessel-Gauss beams, and evolves towards the on-axis crossing point (without considering refraction in the axicon). It can be seen that for higher-order Bessel-Gauss beams the amplitude variation on the annulus maps into an on-axis intensity variation. Additionally, the length of this on-axis, modulated depth-of-field increases with radial order m. Specifically, its range can be approximated as Δz ∼ w_m/ sin (ε₀) [2], where w_m stands for the effective mode size of the m-th order Hermite-Gaussian, which is related to the Gaussian waist as $w_{m} \approx \sqrt{m} \cdot w$ [23]. For a generalized Bessel-Gaussian beam, we find an approximate expression by evaluating the Gaussian waist at the geometrical on-axis crossing point z_c = r_d₀/ tan (ε₀), i.e. w = w₀ · (1+(z_c/z_R)²)^(1/2), and for an ordinary Bessel-Gauss beam w = w₀ since z_c = 0. To adjust the on-axis intensity for laser applications, several parameters of the azimuthally symmetric higher-order Bessel-Gauss beams can be adjusted. Key beam parameters are the mode order m, the waist w₀, the semi-aperture angle ε and the ring radius at minimum waist r_d0.

Fig. 7 (a) Absolute value of the amplitude of the generalized Bessel-Gauss beam vs distance from the minimum Gaussian waist point, (b) and (c): patterns for the (m=1) and (m=2) azimuthally symmetric higher-order Bessel-Gauss beams, respectively.

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3.4. Experimental setup

To observe the higher-order Bessel-Gauss beams experimentally, we have designed an enhancement cavity according to the guidelines, which have been presented in section 3.1. As shown in Fig. 5, the resonator is formed by an axicon mirror (AM) and a spherical mirror (SM). In the experiment, the conic side of the axicon mirror has a base angle of 0.5° and the coating has a reflectance of 0.99. The spherical mirror has a radius of curvature of R = 250 mm and a reflectance of 0.9. The 1-inch diameter spherical mirror (with a 1/4-inch thickness of the BK-7 substrate) serves as the input coupler to the resonator. The back sides of both mirrors are anti-reflection coated. In the design configuration, the separation distance between the two mirrors is L = 75 mm, and the ring radius of the Bessel-Gauss mode on the axicon is approximately 1.5 mm.

A schematic of the excitation path to the resonator is shown in Fig. 8. We use a tunable grating-stabilized external cavity diode laser, which is coupled to a single-mode fiber (PM980-XP, mode-field diameter of approximately 6.6 μm). The laser is operated at a wavelength of 1040 nm and it emits up to 30 mW of power at the fiber output with a line width of 100 kHz. The laser frequency is varied by tuning the grating angle of the external cavity with a piezoelectric element. This allows us to lock the laser frequency to the resonance of the enhancement cavity by dither locking. For this, the light that is reflected from the in-coupling mirror SM is detected at the side port of an optical isolator with a photodiode and is used as a feedback signal for the locking loop. The leakage through the axicon mirror is used to record the resonator modes with a camera, which is placed behind AM.

Fig. 8 Schematic of the experimental setup.

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To match the laser beam to the resonator mode, we first magnify the beam emanating from the fiber by a factor of 450 using the lenses L1, L2, and L3 with focal lengths of 4 mm, 10 mm, 100 mm, respectively. An axicon (A) generates an approximate Bessel-Gauss beam with a semi-aperture angle of (n − 1) · δ = 0.45°, where n is the refractive index of fused silica (n=1.45) and δ the base angle of the axicon, δ = 1°. The waist of the beam illuminating the axicon is 1.5 mm. Due to the round, and thus imperfect, apex of the axicon, the resulting beam in the far field is not a pure ring-like beam but still contains unwanted light in the central region. The quality of the beam is increased by employing Fourier filtering with an opaque disk of 1 mm radius in a 4-f system (150 mm focal-length lenses L4 and L5) [24]. Behind the 4-f system, a 300 mm focal-length lens (L6) and a 200 mm focal-length lens (L7) match the excitation beam to the beam parameters required at the plane side of SM.

To determine the positions of the lenses L6 and L7, at first, we consider a configuration that assumes an ideal Bessel-Gauss beam being produced by the axicon. After the 4-f system the excitation beam is collimated by the lens L6 and is then focused into the resonator by the lens L7. This results in a semi-aperture angle of 3/2· 0.45° = 0.68°. The ring radius of the beam on SM determines the distance between L7 and SM. To better match the beam to the cavity mode, the excitation path is simulated numerically. We account for an imperfect axicon showing a parabolic shape that deviates by 8 μm from the ideal conic shape at the tip. As a consequence, we iteratively improve the distances between L5 and L6, L6 and L7, as well as L7 and SM for best excitation. The resulting values are given in parentheses in Fig. 8, and are used as a starting point for the experiment, in which we also adjust the resonator mirrors and set the final positions of L6 and L7 so that the photodiode signal shows the most pronounced resonance feature.

3.5. Experimental results: comparison of the experimental modes to the beam solutions

Figure 9 shows the photodiode signal during a scan of the laser frequency. The resonances of the higher-order modes are much less pronounced compared to the one of the fundamental Bessel-Gauss mode. This is due to the lower excitation efficiency of the higher-order beam with the beam coupled into the cavity. However, the resonance can be still recognized in the photodiode signal and images of the resonator modes can be taken, as shown in Fig. 10(a)–(d). To obtain these pictures, we limit the range of the frequency scan to the resonance region of the mode of interest. The images are recorded at a scan rate of the piezo that is fast compared to the integration time of the camera. In this way, the camera averages several scans. These images were taken at a distance z= 25 mm behind the flat surface of the axicon mirror. The evolution of the beam profile as a function of other distances from the axicon mirror will be similar to the one shown in Fig. 7. However, it must be noted that in Fig. 7 we have neglected refraction effects by the axicon-mirror substrate. Behind the axicon the semi-aperture angle of the higher-order Bessel-Gauss mode is changed from 0.5° to 0.725° due to refraction. Compared to Fig. 7, this will move the on-axis crossing-point of the beam closer to the axicon mirror. In Fig. 10 azimuthal variations of the intensity on the annulus can be seen, which are due to surface imperfections of the resonator mirrors [11]. Faint central rings can be observed in the experimental data for the higher-order modes, Fig. 10 (b), 10(c) and 10(d). These artifacts are due to transmitted excitation light. The main reason for the show up are the weak resonances of the higher order modes compared to the fundamental mode, as shown in Fig. 9. Figures 10(e)–(h) shows numerical simulations corresponding to the experimental data (taking the refraction of the beam in the axicon mirror substrate into account).

Fig. 9 Signal of the photodiode. The labels correspond to the mode images in Fig. 10.

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Fig. 10 (a)–(d): Experimentally obtained images of higher-order modes m= 0, 1, 2, 3, respectively. (e)–(h): corresponding numerical simulations.

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4. Conclusion

In this work, Bessel-Gauss beams have been generalized to include a set of higher-order solutions that are analytically derived and experimentally demonstrated. Expressions for these beams have been obtained by superposing decentered Hermite-Gaussian beams. The well-known Bessel-Gauss beam is the fundamental solution of this class. Moreover, we have rigorously demonstrated that the azimuthally symmetric Laguerre-Gaussian beams are special cases of these higher-order Bessel-Gauss beams. Thus, these formerly distinct beam types i.e. Bessel-Gauss beams and Laguerre-Gauss beams have been unified. Furthermore, we have shown that the generalized higher-order Bessel-Gauss beams are present as higher-order modes in optical resonators consisting of aspheric mirrors.

Enhancement cavities based on Bessel-Gauss beams are seen to be an increasingly important way to employ strong-field physics in practical applications. Higher-order Bessel-Gauss beams can be used to tailor the intensity distribution in such devices. Other fields of application for these beams are imaging or control and manipulation of particles.

Appendix

To solve the integral of Eq. (12), we write the Hermite-polynomials H_m in their explicit form. For this we use their recursion relation: H₀(t) = 1, H₁(t) = 2t and H_m₊₁(t) = 2tH_m(t) − 2mH_m₋₁(t). This results in an expansions in powers of cos-functions. The n-th power of the cos-functions can be written in terms of the harmonics of its argument by using De Moivre’s formula, e.g.:

\begin{array}{l} {cos}^{2} (β) = (1 + cos (2 β)) / 2 \\ {cos}^{3} (β) = (3 cos (β) + cos (3 β)) / 4 \\ {cos}^{4} (β) = (3 + 4 cos (2 β) + cos (4 β)) / 8 \end{array}

Then, we make use of the following relation

\frac{e^{i l θ}}{2 π} \int_{0}^{2 π} d β e^{i α cos β} cos (n β) e^{- i l β} = \frac{e^{i l θ} \cdot i^{n + l}}{2} [{(- 1)}^{l} J_{n - l} (α) + J_{n + 1} (α)],

which can be easily shown by writing the cosine function in terms of exponentials and applying the definition of the Bessel-function, e.g. as given in [12]:

J_{k} (t) = \frac{1}{2 π i^{k}} \int_{0}^{2 π} d β exp {i (t cos β - k β)} .

Moreover, we use the relation J_−k(t) = (−1)^k · J_k(t).

Acknowledgments

This work is supported by AFOSR grant FA9550-10-1-0063 and the Center for Free-Electron Laser Science. William P. Putnam acknowledges support by a NSF Graduate Fellowship.

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Generalizing higher-order Bessel-Gauss beams: analytical description and demonstration

Abstract

1. Introduction