Quasi-Bessel beams from asymmetric and astigmatic illumination sources

Angelina Müller; Matthias C. Wapler; Ulrich T. Schwarz; Markus Reisacher; Katarzyna Holc; Oliver Ambacher; Ulrike Wallrabe

doi:10.1364/OE.24.017433

1. Introduction

So-called diffraction-free light beams were first described by Durnin et al. more than 25 years ago [1, 2]. Since then, more types of quasi propagation invariant beams (QPIB) have been introduced with many different applications, for example in non-linear optics [3–6] or optical imaging, in optical coherence tomography [7], optical tweezers [8] or for light sheet microscopy [9]. QPIBs can be generated for example by spatial light modulators, which are also used to replace other optical elements such as gratings [10, 11]. One specific class of QPIBs are quasi-Bessel beams (QBB), which are generated with conical lenses, as axicons [3, 12, 13] or axicon mirrors [14] or equivalent configurations of spatial light modulators. Self-reconstruction of QBB has been extensively discussed in literature. Bouchal et al. showed that self-reconstruction of QBB is possible with a simple experiment: They placed obstacles of different shapes and sizes in the optical path of a QBB, which then self-reconstructed itself after a shadowed region behind the absorber [15]. Typically, Bessel beams are produced using symmetric illumination sources, e.g. mono-mode fibers. However, in the context of miniaturization and integration, the question arises to which extent QPIBs can be generated directly from laser diodes, how the profile of those beams propagates and whether the self-reconstructing properties are still valid. The far-field intensity profiles of edge emitting laser diodes typically have an elliptical shape and an astigmatism. Furthermore, group-III-nitrides laser diodes in the green to violet spectral region frequently have a characteristic asymmetric side lobe (substrate mode), which is caused by losses in the substrate (see e.g. [16, 17]).

Thus, we investigate in this paper the beam profiles generated by laser diodes with asymmetric intensity profiles using a refractive axicon lens (with a slightly rounded tip) and compare those to a symmetric illumination source (mono-mode fiber). To analyze this aspect in its full extent, we focus on Fresnel numbers larger than 10, where we have relatively large apertures and thus do not suffer from significant diffraction issues. We compare the characteristic properties of the generated beams and evaluate in all regions to which extent self-reconstruction is possible. We further verify our measurements with simulations using fast Fourier transformations (FFT) and discuss and estimate analytically the effects of the ellipticity, astigmatism and the substrate mode in addition to the effects of the rounding of the tip of realistic physical axicons.

This paper is organized as follows: We discuss the theoretical considerations of QBBs with asymmetric illumination, the effects of astigmatism and substrate modes in laser diodes in Section 2 in addition to the influence of the rounding of the axicon tip and the numerical method that we use. In Section 3, we then present our experimental setup, the parameters of the illumination sources and the data analysis. In Section 4, we show and discuss our results: First, we qualitatively discuss the intensity profile along the optical axis of all different illumination sources. Then, we quantitatively compare the characteristic parameters. Finally, we will discuss the self-reconstructive properties, and we summarize and conclude our results in Section 5.

2. Theorical considerations

In the following section, we first review the concept of a perfect QBB and then discuss the effects of the asymmetric intensity profile of the light source, the astigmatism, the substrate mode and the effects of a rounded axicon. Finally, we describe the numerical method that we use to simulate the beam profile.

2.1. Quasi-Bessel beam

A quasi-Bessel beam (QBB) is generated by a coherent, conical wavefront with an angle α between the optical axis and the normal of the wavefront, see Fig. 1. This can be produced, e.g., by an axicon with apex angle γ and refractive index n. The resulting electric field on a transverse plane is described by

ψ (ρ, z) = ψ_{0} (ρ, z) e^{i k_{z} z} J_{0} (κ ρ),

where ρ² = x² + y² is the radial coordinate, k_z the wave vector in propagation direction, κ is the transverse component of the wave vector, and J₀ is the 0^th-order Bessel function of the first kind [1]. The amplitude ψ₀(ρ, z) is slowly varying and depends on the intensity distribution of the beam illuminating the axicon. From the electric field, we get the intensity I ∝ |ψ|²:

I (ρ, z) = I_{0} (ρ, z) J_{0}^{2} (κ ρ) .

One characteristic parameter is the core radius r_core which describes the radius from the center, i.e. the point of highest intensity, to the first minimum:

r_{core} = \frac{j_{1, 0} λ}{2 π n \sin (α)},

where j_1,0 = 2.4048 is the first zero point of the 0^th-order Bessel function and λ the wavelength. α = (n_axicon − n_air)(π − (γ/2)) is the angle of the conically shaped wavefront generated by an axicon with apex angle γ and n the at a given wavelength in the propagation medium, in our case air. The length of the Bessel region in which the self-interference geometrically takes place is given by

z_{\max} = \frac{r_{\max}}{\tan (α)},

where r_max is the radius of the initial intensity profile, in our case we define it as the point where the intensity distribution has dropped to 1/e² of the maximum value. Similarly, any position on the optical axis z corresponds to a radial position r in the initial beam profile, see Fig. 1(a). When illuminated with a hypothetical plane wave of uniform intensity, the axicon leads to a linear intensity rise along the optical axes z [10]. However, when illuminated by a Gaussian beam at z_max/2 the exponential drop in the initial intensity profile takes over and we have a decrease in the intensity [18], as we discuss further in the next section.

Fig. 1 (a) Schematic illustration of an ideal Bessel beam. (b) Illustration of the asymmetric beam profile, axicon and lateral intensity distributions of slow and fast axis region at different positions. The angle β is the opening angle of the bow-tie intensity pattern.

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Edge emitting laser diodes can be collimated using small collimation lenses with small focal length. For that matter, the influence of the Fresnel number needs to be addressed. In particular micro-optical systems with integrated laser diodes often possess small apertures. Therefore, it is important to discuss in which configurations low Fresnel numbers will appear, as mentioned in [19]. The Fresnel number is defined as the ratio between the square of the aperture radius and the length of an optical system together with the wavelength, F = r²/(Lλ). This will be in our case:

F = \frac{{(\tan (θ) f)}^{2}}{z_{\max} λ} = \frac{{(\tan (θ) f)}^{2}}{\frac{\tan (θ) f}{\tan (α)} λ} = \frac{\tan (α) \tan (θ) f}{λ},

where f is the focal length of the collimating lens and θ the half opening angle of the illumination sources. Fraunhofer diffraction can be assumed for Fresnel numbers F ≪ 1. In the regime where F ≫ 1, we can assume low diffraction. To illustrate for which configurations this regimes are valid, we show in Fig. 2 the relation between the length of the system and the focal length of the collimating lens for Fresnel numbers 0.1 and 10, and different typical opening angles. In our case, we have F > 10. For small Fresnel numbers, there will be significant deviations from the Bessel profile of Eq. (2), as discussed in detail in [19].

Fig. 2 Contours of Fresnel numbers F = 0.1 and F = 10 for different typical opening angles (θ = 0.1, 0.2 and 0.3 mrad) in the plane of the focal length and the maximum focal zone z_max, both normalized to the wavelength. The points indicate the fast and slow axes of our laser diodes and the mono-mode fiber as discussed in section 3.2, with the setup of section 3.1.

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2.2. Asymmetric intensity profile

The beam profile of the edge emitting laser diodes that we use is approximately elliptic:

I = I_{0} e^{- 2 θ^{2} (\frac{\cos^{2} (Φ)}{θ_{slow}^{2}} + \frac{\sin^{2} Φ}{θ_{fast}^{2}})},

where the 1/e² half opening angles are θ_slow and θ_fast, slow stands for the shorter and fast for the longer axis in the far field. After collimation with a perfect lens with focal length f, the intensity profile is for sufficiently small opening angles

I \approx I_{0} e^{- 2 r^{2} (\frac{\cos^{2} (Φ)}{r_{slow}^{2}} + \frac{\sin^{2} Φ}{r_{fast}^{2}})},

where r_slow = f tan θ_slow and r_fast = f tan θ_fast. Similar to Eq. (4), we can define two different propagation lengths:

z_{slow} = r_{slow} / \tan (α) and z_{fast} = r_{fast} / \tan (α) .

Furthermore, the laser diodes have an astigmatism, which will be discussed in more detail in Section 2.3.

The consequence of the asymmetric illumination source is a QPIB, where we can distinguish different regions that are illustrated in Fig. 1(b). The slow axis region with z ≪ r_slow/tan(α) corresponds to the approximately rotationally symmetric central region of the initial profile, where we expect a QBB. Beyond this range, from z = r_slow/tan(α) to z = r_fast/tan(α), the corresponding radius of the initial intensity profile is no more rotationally symmetric as a result of azimuthal asymmetry of the intensity distribution on the corresponding radius. This region has a bow-tie intensity pattern similar to the one described in literature as a Mathieu beam [20–22], which is a member of the family of propagation invariant beams similar to the Bessel beam.

The opening angle β, see Fig. 1(b), of this bow-tie intensity pattern can be derived by straightforward geometric considerations. Let us imagine how light from a point on the fast axis, Φ = π/2 and from a point off-axis, Φ = π/2−β/2 with the same radius r propagate from z = 0 through the point on the optical axis z = r/tan(α) and then forms the inner most part of the bow-tie pattern near the axis. Its azimuthal distribution is approximately the same as the corresponding azimuthal distribution at a radius r of the incident beam. Then, we define β as the angle where the intensity drops to 1/e², i.e. we apply the condition I(r,π/2 − β/2)/I(r,π/2) = e⁻² and get:

e^{- 2} = e^{- 2 r^{2} (\frac{\sin^{2} (β / 2)}{r_{slow}^{2}} + \frac{\cos^{2} (β / 2)}{r_{fast}^{2}} - \frac{1}{r_{fast}^{2}})} .

Substituting r = ztan (α), we can solve this expression for β to obtain

β = 2 \arcsin (\frac{1}{z \tan (α) \sqrt{\frac{1}{r_{slow}^{2}} - \frac{1}{r_{fast}^{2}}}}),

which we show in Fig. 3, left as a function of z/z_fast = ztan (α/r_fast). This expression is valid only for sufficiently large values of z as the circular pattern only starts to open up when the argument of the arcsin becomes smaller than 1. This happens around

z ~ 1 / (\tan (α) \sqrt{\frac{1}{r_{slow}^{2}} - \frac{1}{r_{fast}^{2}}})

and the bow-tie shape will eventually turn into a narrow line shape profile. The latter results from the fast axis radius r_fast in the outer part where the initial beam profile has a similar effect as a slit aperture.

Fig. 3 Left: Calculated opening angle β as a function of the ratio z/z_fast for different values r_slow/r_fast. The laser diodes which we use and which are described in section 3.2 have a ratio r_slow/r_fast of 0.4 and 0.5, respectively. Right: Ratio of the z-position of the maximum intensity z_Imax to the slow axis range z_slow as a function of the ratio of the 1/e² radii r_slow/r_fast.

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Next, let us look at the effect that the asymmetry has on the intensity distribution in the central maximum. The intensity profile along the propagation direction z has its maximum at z_max/2 for the case of a symmetric illumination source with Gaussian intensity profile, but for the asymmetric laser diodes this is not necessarily the case.

The intensity of the central maximum at a position z, I_max(z = r/tan (α)), is proportional to the integrated intensity on a circle with the corresponding radius r on the plane of the axicon:

\begin{array}{l} I_{int} (r) = \int_{0}^{2 π r} I d l \\ = r \int_{0}^{2 π} e^{- 2 r^{2}} ((\frac{1}{r_{slow}^{2}} - \cos^{2} (Φ)) + (\frac{1}{r_{fast}^{2}} - \cos^{2} (Φ))) d Φ . \end{array}

After substituting r = ztan (α), computing the integral and simplifying the expression, we get

I_{\max} (z) \propto 2 π z \tan (α) e^{- {(z \tan (α))}^{2} (\frac{1}{r_{slow}^{2}} + \frac{1}{r_{fast}^{2}})} J_{0} ({(z \tan (α))}^{2} (\frac{1}{r_{fast}^{2}} - \frac{1}{r_{slow}^{2}})) .

From this Eq. we can find the numerical value for the ratio of the z-position with maximum intensity to the length of the slow axis range z_Imax/z_slow as a function of the ratio of the initial beam radii r_slow/r_fast as shown in Fig. 3, right. If the initial intensity profile is symmetric, the ratio r_slow/r_fast equals one, so we see that z_Imax = z_slow/2 as expected. In the asymmetric limit of r_slow/r_fast → 0, we get z_Imax = 0.89 z_slow. This effect is much stronger than a similar focal shift of focused elliptic Gaussian beams that is discussed in [23].

2.3. Effects of astigmatism in laser diodes

Edge emitting laser diodes have an astigmatism which is described in literature to be in the region of several micrometers (3–10 µm) [24]. This will introduce asymmetric phase differences and can destroy the interference that leads to the Bessel beam pattern. In this subsection, we will focus on the interference condition for the central maximum and will derive a “safe” condition under which the asymmetric intensity distribution will actually suppress the interference effects from the astigmatism.

The astigmatism means that the origin of the curvature of the wavefront in the slow and fast axes is shifted by a distance a, as shown in Fig. 4. Hence, we can collimate such a wavefront with our collimation lens only in one axis. The other axis still has a curvature in the wavefront, causing a different propagation length ∆l_astigm(r). This implies that the light arriving from the fast and slow axis direction by ray geometry at a point z on the central axis comes from two different radii on the plane of the axicon. Thus, we see a difference in the conical angle ∆α and in the length of the light path, ∆l_cone depending on the azimuthal angle Φ.

Fig. 4 The effects of astigmatism a. The astigmatic wavefront is collimated only in one direction. This generates an angular shift ∆α.

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The condition for constructive interference is that the variation of the optical path is smaller than ∆l = λ/2. Using the expression

l = \frac{z}{\cos (α)},

we can relate the variation in the length of the light path to a variation in the conical angle

Δ l_{cone} = z Δ α \frac{\sin (α)}{\cos^{2} (α)} .

The total shift is then the combination of geometric and astigmatic length shift

\begin{array}{l} Δ l_{tot} = Δ l_{cone} + Δ l_{astigm} \\ = z Δ α \frac{\sin (α)}{\cos^{2} (α)} + \frac{1}{2} \frac{a r^{2}}{f} . \end{array}

At a radial position r on the axicon, a shift a of the origin from the wavefront of a collimated beam results in an angular shift ∆α = (r a)/f ² and if we substitute the radius r = f tan(θ), we get

Δ l_{tot} = a (\frac{1}{2} - \frac{1}{\cos (α)}) \tan^{2} (θ) .

Assuming then a simplified picture where we have just two wave fronts, one from the fast and one from the slow axis, the condition for destructive interference is:

θ_{destructive} = \arctan (\sqrt{\frac{λ \cos (α)}{a \cos (α) - 2}}) .

The relative amplitude of the interference can be roughly estimated by the ratio of initial intensity distributions of slow and fast axis:

\frac{I_{interference}}{I_{total}} \approx \frac{I_{slow}}{I_{fast}} .

Substituting the Gaussian distribution of Eq. (6), we get

\frac{I_{θ |_{Φ = 0}}}{I_{θ |_{Φ = \frac{π}{2}}}} = e^{- 2 θ^{2} (\frac{1}{θ_{slow}^{2}} - \frac{1}{θ_{fast}^{2}})} = e^{- 2 \frac{θ^{2}}{θ_{m}^{2}}}

where θ_m is an effective opening angle that describes the faster decline of the slow axis compared to the fast axis,

θ_{m} = \frac{1}{\sqrt{\frac{1}{θ_{slow}^{2}} - \frac{1}{θ_{fast}^{2}}}} .

In order to have a relative amplitude of the interference effect less than 1/e², we need in our simplified model:

θ_{m} \leq θ_{destructive}

Substituting Eq. (17) and Eq. (20), this becomes

\tan^{2} (θ_{m}) \leq (\frac{λ \cos (α)}{a (\cos (α) - 2)})

and solving for the wavefront angle α, we get

α \leq \arccos (\frac{2 a \tan^{2} (θ_{m})}{λ + a \tan^{2} (θ_{m})}) .

This inequality can further be simplified by introduction of a critical parameter g that only depends on the properties of the laser diode, g = (2atan² (θ_m))/λ:

α \leq α_{critical} = \arccos (\frac{2 g}{2 + g}) .

For angles α of the wavefront smaller than α_critical, or critical parameters g < g_critical = (2cos(α)/(2 − cos(α)) as shown in Fig. 5, left, interference will be highly suppressed, due to the difference in the initial intensity profiles. In fact, the maximum realizable refractive axicon angle with a typical refractive index of n = 1.5 can generate at most an angle α = 30°, see left Fig. 5, so that the smallest relevant critical parameter is g = 1.53. We indicate different astigmatism values a with the effective half opening angle θ_m = 166.2 mrad and λ = 436 nm, which will be relevant in section 3.2. We see that for our case the astigmatism in the laser diodes has no influence on our measurement as we stay within the shaded border.

To validate our estimate, we simulated the astigmatism values indicated in Fig. 5, left, using the method described in section 2.6, for an ideal axicon with apex angle γ = 178° and wavelength λ = 436 nm. In Fig. 5, right, we show the normalized center intensity for symmetric (θ_m = 0) and asymmetric (θ_m = 166.2 mrad) illumination sources with same astigmatism. We clearly see that the symmetric illumination sources have a perfect destructive interference at the center for certain values of z, whereas the interference is suppressed in the asymmetric sources with smaller values of g. The suppression is somewhat smaller than expected from our simple model, as in reality there are not only two wave fronts from the fast and slow axis, but a continuous wavefront with smoothly varying phase and intensity. In the full 3D intensity distribution, we expect that the interference appears in the transverse intensity profile as a loss of contrast around the central maximum.

Fig. 5 Left: The critical angle αcritical as a function of the parameter g = (2atan² (θ_m))/λ. The shaded area indicates the maximum realizable area, where the maximal wavefront angle α = 30° with a refractive index n = 1.5 of the axicon is the physical limitation. The vertical lines are calculated examples of a mean half opening angle θ_m = 166.2 mrad and λ = 436 nm for different astigmatism values. Right: Normalized core intensity of a simulated ideal axicon with apex angle γ = 178° for symmetric (sym) and asymmetric (asym) illumination and different astigmatism values a. The simulations will be discussed in more detail in section 2.6.

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2.4. Effects of substrate modes and far-field ripples in laser diodes

Group-III-nitride edge emitting laser diodes typically use the same material GaN for the waveguide and for the substrate. Consequently, the waveguide mode may partially leak into the substrate, causing substrate modes which can be seen in their initial far-field intensity distribution at some angle θ_s. As the substrate mode extends over the whole thickness of the substrate, its opening angle ∆θ_s is relatively small [17]:

Δ θ_{s} ~ \frac{λ}{d_{s}},

where d_s is the thickness of the substrate as seen in Fig. 6. It has a non-trivial effect on the Bessel beam as it is not collimated with the main Gaussian intensity distribution because its origin is slightly displaced compared to the laser mode. The transverse offset is approximately d_s/2 so that it will have an angle relative to the main beam:

φ ≃ \frac{d_{s}}{2 f} .

Furthermore, the lateral displacement of the substrate mode will give us an opening angle ∆ϕ behind the collimation lens. In addition to substrate modes, additional narrow peaks may arise in the far-field, which are caused by light scattering from geometric ripples off the laser diode waveguide to the cladding interface. These ripple modes are again typical for group-III-nitride edge emitting laser diodes, both because of the relatively short wavelength leading to higher scattering efficiency, and because of epitaxial growth mechanisms leading to an undulation of the growth surface.

Fig. 6 Schematic influence of the substrate modes. The lateral and transverse displacement of the substrate mode will result in a widening in the z-direction ∆z.

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To see at which position z_s on the optical axis the substrate mode will appear, we have to take into account the position of the substrate mode in the plane of the aspherical lens, h_s = f tan(θ_s), which then gives us together with the angle ϕ the position in the plane of the axicon h_a and hence:

z_{s} = \frac{h_{a}}{\tan (α + φ)} = \frac{(h_{s} - l_{a} \tan (φ))}{\tan (α + φ)},

where l_a is the distance between the lens and the axicon. The length over which the substrate mode appears comes both from the opening of the substrate mode itself, ∆h_s, and its defocusing

Δ z ~ \frac{Δ h_{a}}{\tan (α + φ)} + \frac{h - l_{a} \tan (φ)}{\sin^{2} (α + φ)} Δ φ,

where ∆h_a = ∆h_s + l_a tan(φ). Because of the defocusing, ripples and substrate modes will run into each other such that we consider all of them for ∆h_s. As the substrate mode is coherent with the laser mode, it can interfere with the main beam. To estimate this effect, we have to compute the length difference of the corresponding optical paths, where we will ignore any fixed phase difference between the modes:

Δ L = L_{beam} - L_{s} = \frac{z}{\cos (α)} - \sqrt{h_{a}^{2} + z^{2}} .

The condition for constructive interference ∆L = nλ, n ∈ Z will then be

\frac{z {(α + φ)}^{2}}{2} - \frac{h_{a}^{2}}{z} = n λ .

For typical laser diodes and not extremely small axicon angles α, n will be of the order of at least a few hundred, e.g. n = 150 for λ = 431 nm and γ = 178° (h_a = 4.7 mm,α + φ = 0.46°). Hence, we will have a very rapidly varying interference pattern in the z-direction where the maxima might not even be distinguishable. For even larger n, i.e. not small angles α, the path difference will be greater than the coherence length of the laser diode so that no interference effects will be visible, e.g. n = 448 (∆L = 195m) for γ = 160°. Furthermore, the ripples running into the substrate mode may cause decoherence so that we do not have interference in the first place.

2.5. Rounding of the axicon tip

The ideal axicon has a perfectly sharp tip. In reality, the tip of actual refractive axicon lenses is rounded and has a significant influence in the generation of the QBB. This results in a periodic intensity modulation described in [13] and can be explained by the interference of two waves, as illustrated in Fig. 7: We assume a plane wavefront from the axicon tip with a phase ϕ₁ = kz and a conical wavefront with ϕ₂ = kz/cos(α). If we have constructive interference, we can write,

| ϕ_{2} - ϕ_{1} | = 2 π n, n \in Z

and substituting ϕ₁ and ϕ₂:

k z_{n} (\frac{1}{\cos (α)} - 1) = 2 π n .

From this expression we get the modulation period in propagation direction, for small angles

z_{n} ≃ \frac{2 λ n}{α^{2}},

where k is the wave vector and α is again the conical angle of the wavefront. Another important issue of the rounded tip is that the rounding acts as a lens with a radius of curvature r_c (see section 3.1) and a focal length z₀ = r_c/(n − 1) which is illustrated in Fig. 7. Before z₀, no Bessel beam can be generated if we assume that we have a continuous sufficiently smooth transition from the rounding into the conical axicon slope.

Fig. 7 Schematic of the two wave fronts: the plane wave ϕ₁ and the conical wave ϕ₂ with indications of the tip rounding and the focal length z₀.

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A second intensity variation due to the rounding of the axicon tip can be observed at the first minimum of the beam profile where an interference of the “plane wave” of the axicon tip and the conical wave cause a shift and modulation of the minimum. The approximate electric field from Eq. (1) near the minimum r_core of the Bessel beam is

Ψ_{Bessel} = Ψ_{0, Bessel} (z) e^{i k_{z} z} κ Δ ρ J_{0}^{'} (j_{1, 0}),

where ∆ρ = ρ − r_core and

J_{0}^{'} (j_{1, 0})

is the derivative of the 0^th-order Bessel function at its first zero point. Further, we assume the electric field from the rounded tip to be

Ψ_{lens} = Ψ_{0, lens} (z) e^{i k z} .

If those two electric fields interfere, we can write

\begin{array}{l} Ψ_{overall} = Ψ_{Bessel} + Ψ_{lens} \\ = Ψ_{0, lens} e^{i k z} (1 + \frac{Ψ_{0, axicon}}{Ψ_{0, lens}} e^{^{i} (k_{z} - k) z} (κ Δ ρ J_{0}^{'} (j_{1, 0}))) . \end{array}

After simplification, the intensity distribution I ∝ |Ψ|² becomes

I_{overall} \propto 1 + 2 \frac{Ψ_{0, axicon}}{Ψ_{0, lens}} κ Δ ρ J_{0}^{'} (j_{1, 0}) \cos^{2} ((k_{z} - k) z) + {(\frac{Ψ_{0, axicon}}{Ψ_{0, lens}} κ Δ ρ J_{0}^{'} (j_{1, 0}))}^{2} .

The shift of the minimum ∆ρ_min is then:

Δ ρ_{\min} = - \frac{Ψ_{0, axicon}}{Ψ_{0, lens}} \frac{1}{κ J_{0}^{'} (j_{1, 0})} \cos ((k_{z} - k) z),

i.e. we get the same modulation period in the propagation direction z_n = ((2λn)/α²) as obtained in Eq. (33). The amplitude of this oscillation, i.e. the amount by which the first zero point varies, is

(Ψ_{0, axicon} / Ψ_{0, lens}) (λ / α 2 π J_{0}^{'} (j_{1, 0}))

, so it is proportional to the ratio

\sqrt{(I_{Bassel} / I_{lens})}

and by Eq. (3) to r_core. If we now substitute ∆ρ_min back into the overall electric field, Eq. (37), we find that the amplitude actually only vanishes at discrete values where cos ((k_z − k)z) = ±1, provided that our linearization is consistent.

2.6. Simulations

To gain more insight and verify our measurements, we have numerically simulated the beam propagation using fast Fourier transformations (FFT) ℱ (·). To do so, we adapted the procedure developed for the rotationally symmetric case by Siegman, which uses a quasi fast Hankel transformation to evaluate the wave function [25]. This yields the propagation of an initial wave Ψ_initial up to a point z:

Ψ (z) = ℱ^{- 1} (P (r, z) ℱ (Ψ_{initial}))

with a propagator

P (r, z) = e^{- i z} \sqrt{k^{2} - {(κ_{x}^{2} + κ_{y}^{2})}^{2}},

where κ is the dual spatial base vector of the x, y - plane. The intensity is proportional to |Ψ|². Assuming a thin lens, the initial wave front was designed to take into account the actually measured shape of the axicon, including the rounding near the tip. The overall electric field behind the axicon at zero z-position is then described by

Ψ_{initial} (r, z = 0) = \sqrt{I_{Gauss} (r)} e^{- i k (n_{ari} - n_{axicon}) Δ z (r)}

where ∆z(r) is the surface profile of the axicon, which will be discussed in more detail in section 3.1. This wavefront is propagated along the optical axis to obtain the QPIB.

To evaluate the self-reconstruction properties of the beams, we performed the simulation in two steps. First, we propagated the wave function to the plane of the absorber on the optical axis. There, we multiplied the wave with an absorber matrix B that represents the shading by the absorber. Finally, we propagated the modified wavefront to the image plane, such that we get overall:

Ψ (z_{image}) = ℱ^{- 1} (P (z_{image} - z_{absorber}) ℱ (B ℱ^{- 1} (P (z_{absorber} - z_{initial}) ℱ (Ψ_{initial})))) .

3. Experimental considerations

We performed the measurements that we describe in this section by a straightforward experimental configuration consisting of an aspherical lens for collimation, an axicon and a CMOS camera. We obtained the parameters for the simulation by characterizing the different illumination sources. Finally, we developed the data analysis to compare the characteristic parameters of the generated QPIBs.

3.1. Experimental setup

To measure the beam profile, we used the experimental setup shown in Fig. 8. As illumination sources we chose a symmetric mono-mode fiber and two different edge emitting laser diodes. Furthermore, we used a refractive anti-reflection coated, fused silica axicon with a nominal apex angle of γ = 178° ± 0.5° to generate the QPIB. In order to perform a realistic simulation, we obtained its actual geometry and the radius of curvature of the tip by measuring its profile with a white light interferometer and fit a general polynomial to the data:

f = \sqrt[2 n]{a_{0} + a_{2} r^{2} + a_{4} r^{4} + \dots + a_{2 n} r^{2 n}} .

We find for our case that the third order polynomial parametrized as,

Δ z (r) = \sqrt[6]{a r^{2} + b r^{4} + c^{6} + \frac{r^{6}}{\tan^{6} (γ / 2)}},

is sufficient to have a good agreement with our measured data. The parameters a, b and c describe the deviation from a perfect axicon and γ is its apex angle. We used a weighting of the data points of

1 / \sqrt{{(100 m)}^{2} + r^{2}}

to weigh the radial sections equally and at the same time cut off the weighting factor to avoid excessive noise from the few data points with diverging weight at the center. The axicons apex angle is γ = (178.553 ± 8 × 10⁻³)°, which lies in the range of the manufacturers. The rounding of the axicon tip corresponds to a small lens with a radius of curvature r_c = 48.84 ± 0.085 mm and a half diameter r_lens = 852.5 ± 0.04 m. This small lens will generate a focal point at z₀ ≈ 105 mm depending on the wavelength as listed in section 3.2, Table 1, only behind which a QBB can be detected. This is the reason why we start our measurements only shortly before z₀ at a z-position of 80 mm.

Fig. 8 Experimental setup consisting of (left to right) the illumination source, the aspherical collimation lens, the axicon and a movable CMOS camera. An optional absorber can be placed at an arbitrary position in the optical path.

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Table 1. Different illumination sources with characteristic parameters.

View Table

To avoid the use of a telescopic setup, we collimated those sources with an aspherical lens with focal length f = 17.5 mm and NA = 0.71. This gives us two advantages: First, we have an overall length of the QPIB that we can conveniently measure on an optical table. Second, we can capture the whole intensity profile of the laser diode. To obtain the intensity profile, we used a transversely (z-direction along the optical axis) movable 5 megapixel CMOS camera with a pixel pitch of 2.2m and a travel range of 500 mm. This camera is able to resolve the core radius of our chosen illumination sources which is around 21m, compare Table 1, without magnification. To evaluate the self-reconstructing properties of the beam, we laser-structured an optional obstacle with diameter 750 ± 4 m out of black paper which can be placed at an arbitrary position in the optical path.

3.2. Illumination source

The initial beam profiles of the illumination sources were measured in the far-field using a CMOS camera with 4 megapixel and a pixel pitch of 5.5 m. We obtain the 1/e² half opening angle of the Gaussian profile from a fit as described in Eq. (7) and calculated the core radius r_core and the length of the QBB region z_max from Eq. (3) and Eq. (4), respectively. We used the refractive index of our axicon n_axicon and the wavelength λ as listed in Table 1. The illumination sources are the symmetric pig-tailed mono mode fiber with a circular Gaussian intensity distribution and the two laser diodes (LD1, LD2). The latter are asymmetric gallium-nitride based, c-plane ridge waveguide with an elliptic Gaussian intensity distribution.

The far-field intensity profiles of the laser diodes are shown in Fig. 9. We used a pulse length of 1s and a duty cycle of 1% to drive the laser diodes with driving currents of I_LD1 = 200 mA and I_LD2 = 150 mA, just above the threshold current.

Fig. 9 Far-field intensity profile of LD1, λ = 431 nm (left) and LD2, λ = 436 nm (right) with indications for the 1/e² drop. Intensity profiles of the fast and slow axes (LD1 center, top and LD2 center, bottom) with markings at characteristic points. The region affected by the rounding of the axicon tip is indicated by the shaded area.

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3.3. Data analysis

To compensate any possible unwanted lateral motion of the camera stage, we analyzed the beam profile in individual lateral planes. We obtained the optical center of gravity and then divided the image in equidistant polar coordinate areas of ∆Φ = 11.25°. To describe the beam profile, we analyzed the following characteristic parameters by fitting a polynomial of 4^th-order into the data in a region of 15m around the minima and maxima in each polar interval: The core intensity I_core, the radius of the first minimum r_core and the intensity of the first maximum I_max,1, all of them as a function of the distance z. The intensities are normalized to the total integrated intensity of the initial intensity distribution measured after the collimation lens without an axicon. As the illumination sources have slightly different wavelengths, we normalized the core radius with the wavelength in order to be able to compare the data. To study the asymmetric behavior of the beam profile, we used the following expression for the ellipticity of the intensity in the first maximum I_max,1 as a measure for the transition from a circular profile to a line-shaped profile:

ε = \frac{(I_{fast, 1} - I_{slow, 1})}{(I_{fast, 1} + I_{slow, 1})},

ε = 0 corresponds to a perfectly symmetric beam and ε = 1 to an asymmetric beam with vanishing intensity in the slow axis direction. We obtained this parameter experimentally by fitting a sinusoidal curve to the values I_max,1(∆Φ).

4. Results and discussion

In this section, we evaluate our measurements and simulations by comparing the beam parameters that we obtain from the measurements to the simulations and also, where possible, to the analytical results and qualitative considerations. To avoid artifacts from the data analysis, we run the same data analysis on the measured intensity distributions and on the simulation results. First, we give a short qualitative overview of the beam profile. Then, we compare the behavior of the core intensity and the core radius with the theoretical value and discuss the asymmetry in the different regions on the optical axis. Finally, we discuss the self-reconstruction properties of the different illumination sources.

4.1. QPIB profiles

We reconstructed the spatial intensity pattern obtained from simulations in the planes of the fast and slow axis as shown in the left part of Fig. 10. The intensity is scaled logarithmically for better illustration of the side lobes. On the right, we show the lateral cross-sections (normalized to the brightest pixel) at different positions (1–4) and compare the simulations (top) to the measurements (bottom). As the simulation and also approximately the measurement were illuminated with a Gaussian intensity distribution of Eq. (7) and Fig. 9, respectively, we expect a Bessel-Gaussian beam type, which is described for symmetric beams in the literature [3,13,18]. For the transverse intensity images we chose special points along the axis where the transition from a symmetric QBB into a bow-tie QPIB can be seen: (B,C 1) z = 125 mm (behind z₀, but in the slow axis region), (B,C 2) the end of the slow axis region z_max,slow, (B,C 3) an intermediate position $z_{mean} = \sqrt{z_{\max, slow} z_{\max, fast}}$ and (B,C 4) the end of the fast axis region z_max,fast. The simulation of the fiber is shown only in one plane due to rotational symmetry. The symmetric interference pattern is found in all three cross-sections, (A1) z, (A2) z_max,(A3) 1.5 z_max. The optical properties of the used measurement setup have a negligible influence on the quality of the beams as can bee seen for the fiber in Fig. 10, top. Here, we can also see that the measurements are in good agreement with the simulations, up to some loss of contrast of the varying intensities of the rings.

Fig. 10 Left: Simulations of the QPIB profiles along the optical axis in the plane of the slow and fast axis of the fiber (top), LD1 (center) and LD2 (bottom). To illustrate the side lobes the intensities were logarithmically scaled. Right: Simulation and measurement at different positions of the transverse plane: (A1) z = 125 mm, (A2) z_max, (A3) 1.5 z_max for the fiber and (B,C 1) z = 125 mm, (B,C 2) z_max,slow, (B,C 3) $z_{mean} = \sqrt{z_{\max, slow} z_{\max, fast}}$ and (B,C 4) z_max,fast for the laser diodes. The images are each normalized to the brightest pixel.

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The QPIBs of the laser diodes start symmetrically in the slow axis region. At z_max,slow, we see a first indication of asymmetry. Near z_mean, a bow-tie beam can be seen and eventually, in the simulations, far outside at z_max,fast, there is almost a line profile. At large distances starting from z_mean, there appear deviations in the experiment, which may be due to the substrate modes that deviate from the simulated Gaussian intensity distribution. This can be seen in Fig. 9, where the initial intensity profile is less smooth in the outer region. In particular the last cross section B4 and C4 in Fig. 10 the experiments differ due to ripples, side reflections or other optical artifacts, which can be seen near θ_max in Fig. 9.

Next, let us compare the measurements of the normalized core intensity I_core with the simulation of a rounded axicon as seen in Fig. 11. We clearly see that the lens effect mentioned in section 2.5, created by the rounding of the tip [shaded area in Fig. 9], generates a focal point at z₀. The QPIB starts at this point, with decreasing intensity as expected for a Gaussian intensity distribution z₀ ≈ z_Imax. We see for all light sources, that the simulation results of the rounded axicon display the oscillation described in section 2.5 and converge to the intensity of a perfect axicon. In fact, we verified that the period agrees with Eq. (33). Overall, the measurements are in reasonably good agreement with the simulations.

Fig. 11 Core intensity of the simulated and measured quasi propagation invariant beam profile along the optical axis for LD1 (top left), LD2 (bottom left) and for the fiber (right).

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For laser diode LD1, we see as the most prominent deviation the peak at z = 300 mm in the measured data, which probably results from the substrate mode described in section 2.4. This position agrees well with the angle ϕ ∼ 2.75 mrad that we measured from the tilt of the substrate mode in the collimated beam, see Fig. 9. This in turn is consistent with a substrate thickness d_substrate ∼ 96 m. The oscillation that we see on the top of the peak and in the tail at z > 400 mm might come from the interference effects. The width ∆z ~ 110 mm is consistent with an opening angle ∆φ ∼ 0.25 mrad that we measured and a ∆h_s ∼ 1 mm, which includes the substrate mode together with the ripples. The same applies qualitatively for laser diode LD2, where we see a peak at 280 mm, an angle φ ∼ 3.8 mrad and a width over which the substrate mode will appear of ∆z ~ 95 mm.

For the symmetric fiber, we find a good agreement between measurement and simulation, compare Fig. 11, right. The slower decrease in the measurement compared to the simulation after z_max can be explained by the outer stray light in the initial intensity profile of the measurement while the simulation assumed a Gaussian intensity profile.

The behavior of the other characteristic parameters, the normalized core radius (r_core) and the ellipticity of all illumination sources as a function of the z-position in measurements and simulations, are shown in Fig. 12. The normalized radii are taken from the first minimum of the original data, averaged over the polar angle as described in section 3.3. We can see a good agreement of measurement and simulation and how the value approaches the value of an ideal axicon r_ideal = 47.25 m after the focal length. Only after the end of the Bessel region z_max,slow, we can see a widening well inside the fast axis region, which may come from the asymmetry. By detailed simulations we verified at some examples that the oscillation which we see here has a period of z_n ≃ (2λn)/α², as predicted by Eq. (33) and hence, this oscillation is an effect of the rounding of the axicon mentioned in section 2.5.

Fig. 12 Measurement and simulation of the core radius r_core (left). Ellipticity in the intensity of the first maximum, see Eq. (45), of the measured and simulated quasi propagation invariant beam profiles (right). For the fiber (λ = 473 nm), LD1 (λ = 431 nm) and LD2 (λ = 436 nm), with indications of the characteristic positions on the optical axis.

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The ellipticity ε from Eq. (45), was introduced to study the transition from a circle (ε = 0) towards a line profile (ε = 1). For the symmetric fiber, the ellipticity stays nearly constant at a value around zero until we see probably the effects of a small signal to noise ratio or side reflections or side scattering at several multiples of z_max. For the laser diodes, ε starts rising at z₀ as the intensity on the slow axis decreases already at that point, while the intensity on the fast axis is still approximately constant. The deviations from measurement and simulation can be explained by the variations of the initial intensity profile and an overall loss of contrast due to scattering, misalignment and reflections, which puts more light into the theoretically dark regions. This effect can also be seen in the cross sectional images (B,C 3) and (B,C 4) in Fig. 10.

4.2. Self-reconstruction

To study the self-reconstruction of the QPIBs, we placed an absorber with diameter d_absorber = 750 ± 4 m at the center of the beam in the setup shown in Fig. 8. By Eq. (4), we can calculate the distance z_disturb ≈ 46 mm (depending on the wavelength because of the chromatic dispersion of the axicon) after which the beam is geometrically reconstructed. The self-reconstruction of the intensity in the central maximum of the different illumination sources is shown in Fig. 13, where we plot the simulated (left) and measured (right) intensity profile along the optical axis. To illustrate the self-reconstruction in the different characteristic regions, we chose the positions of the absorber to be near the characteristic points: A1 = 125 mm, A2 ≈ z_max,slow, $A 3 \approx \sqrt{z_{\max, slow} z_{\max, fast}}$ and A4 ≈ z_max,fast. Taking into account the intensity profile of the unobstructed beam, the intensity after the self-reconstruction of the beam is in good agreement with the theoretical value. This can be seen in all regions throughout the propagation direction, except for the absorber at the end of the fast axis region. In this case, the intensity is already of the order of noise in the measurements and we have the distortions seen in Fig. 10, right. The main difference between simulations and measurements is the smaller slope of the intensity profile near z_disturb. This results probably from the stronger diffraction at the edge of the laser-structured absorber compared to the perfect disk in the simulations.

Fig. 13 Core intensity of the simulated (left) and measured (right) beam profile along the optical axis (z-direction) with an absorber of 750 4 m diameter placed at different positions on the optical axis, for the fiber with λ = 473 nm±(top) at A1 = 125 mm, A2 = z_max and at A3 = 1.5 z_max. The absorber positions for LD1 (λ = 431 nm) (center) are at B1 = 125 mm, B2 = 246 mm, B3 = 356 mm and B4 = 514 mm. For LD2 (λ = 436 nm) (bottom) the positions are C1 = 125 mm, C2 = 276 mm, C3 = 382 mm and C4 = 546 mm. The shaded regions behind the absorbers indicate the distance after which the beam is geometrically reconstructed.

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5. Summary and conclusions

We showed that it is possible to generate quasi propagation invariant beams (QPIB) with non-perfect, asymmetric illumination sources, such as edge emitting laser diodes, together with refractive axicons that have a rounded tip. We found that the asymmetry results in a transition of a symmetric Bessel beam into a bow-tie shaped pattern. We compared the characteristic properties of the generated beams and evaluated in measurements and simulations to which extent self-reconstruction is possible.

In section 2, we studied several characteristic effects analytically. We discussed the effect of asymmetric intensity profiles from laser diodes, where we estimated the opening angle of the bow-tie pattern and showed that the maximum of the intensity for the asymmetric beam shifts along the optical axis compared to the symmetric one. Furthermore, we showed that astigmatism may destroy the formation of a QPIB. However, we also found that this effect is highly suppressed in realistic asymmetric light sources with realistic values of the astigmatism (a = 3 – 10 m) as observed for edge emitting laser diodes and an asymmetric intensity distribution. Another effect that we studied, is the influence of substrate modes in laser diodes. Those are not collimated and eventually interfere with the main beam. Also, we found that the rounded axicon tip causes a periodic intensity modulation as described in [13] and a similar modulation of the core diameter. In section 3 we explained the experimental setup and discussed how we obtained the relevant data for the simulations. In section 4, we found reasonably good agreement between simulations and experiments of QPIBs generated by the laser diodes. Those beams consist of two characteristic regions: First, the approximately isotropic central part of the intensity distribution of the laser diode, which creates a quasi-Bessel beam in the slow-axis region behind the axicon. This turns smoothly into an asymmetric bow-tie shaped beam cross section in the fast axis region illuminated by the outer highly anisotropic part of the intensity profile. The measurement results agree very well with the simulations, the main difference being some variations in the intensity of the central maximum. We showed that these may be explained by the substrate mode which is not collimated together with the main beam. Furthermore, some of the deviations may be due to the fact that we assumed a perfect asymmetric Gaussian profile for the illumination sources while the intensity distribution of the laser diodes does not have such a smooth intensity distribution.

Similarly, we also showed good agreement between the asymmetry of the initial beam and the transition from a quasi-Bessel beam to a bow-tie shaped beam. Still, the existence of the central maximum is robust and not disturbed by such imperfections as we showed in section 4.2. Finally, we verified that the self-reconstruction takes place reliably in all regions of the QPIB.

Acknowledgments

The authors highly appreciate the help of Dr. Andreas Greiner with the computational infrastructure. This work was supported by the BrainLinks-BrainTools Cluster of Excellence funded by the German Research Foundation (DFG, grant no. EXC 1086).

References and links

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Parameters	Fiber	LD1	LD2
λ	473 nm	431 nm	436 nm
n_axicon	1.4639	1.4671	1.4668
r_core	22.35 m	20.23 m	20.48 m
θ_slow	85.5 ± 0.05 mrad	106.8 ± 0.03 mrad	143.8 ± 0.02 mrad
θ_fast		261.1 ± 0.09 mrad	286.4 ± 0.03 mrad
θ_substrate		240.8 mrad	271.6 mrad
z₀	105.3 ± 0.2 mm	104.57 ± 0.2 mm	104.63 ± 0.2 mm
z_max,slow	185.2 ± 0.1 mm	230 ± 0.07 mm	310.8 ± 0.03 mm
z_max,fast		573.6 ± 0.2 mm	632.4 ± 0.07 mm

Quasi-Bessel beams from asymmetric and astigmatic illumination sources

Abstract

1. Introduction

2. Theorical considerations

2.1. Quasi-Bessel beam

2.2. Asymmetric intensity profile

2.3. Effects of astigmatism in laser diodes

2.4. Effects of substrate modes and far-field ripples in laser diodes

2.5. Rounding of the axicon tip

2.6. Simulations

3. Experimental considerations

3.1. Experimental setup

3.2. Illumination source

3.3. Data analysis

4. Results and discussion

4.1. QPIB profiles

4.2. Self-reconstruction

5. Summary and conclusions

Acknowledgments

References and links

Cited By

Figures (13)

Tables (1)

Equations (45)

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