The photonic crystal fiber [1, 2] (PCF) offers a unique opportunity for realizing broadband large-mode area (LMA) single-mode (SM) fibers [3]. In high-power laser applications the LMA property is desirable to keep the power density below the damage threshold of silica as well as to avoid non-linear phenomena and the SM property is essential for the beam quality and the ability to generate close to diffraction-limited beams. Recent studies of LMA-SM-PCF lasers have suggested that the PCF technology may indeed offer good beam quality [4] as quantified by an M-squared value [5] close to unity (M ²=1 for a Gaussian beam).

For the fundamental mode in PCFs the near and far fields have intensity distributions which are overall close to Gaussian, but with a six-fold rotational symmetry reflecting the symmetry of the air-hole lattice. For LMA-PCFs there are no significant deviations from a Gaussian distribution for intensities above the 1/e² level [6] and for many practical applications the mode may be considered Gaussian. However, as studied recently the derivation from a Gaussian distribution has some important consequences [7]. In the near-field the intensity is suppressed at the locations of the air-holes and in the asymptotic far-field these suppressions manifest themselves as low-intensity satellite peaks. Furthermore, in the near field to far-field transition the intensity becomes strongly focused at a finite distance from the end-facet as can be understood by a simple Gaussian decomposition of the near field.

While the simple decomposition approach has proved very useful for the qualitative understanding of the near to far-field transition the laser applications call for a more quantitative study of the radiated field. Calculating the diffraction pattern is generally a complicated task, which involves the full solution to the elastic scattering problem at the end-facet. Here, we rigorously calculate radiated fields from the end-facet utilizing a surface equivalence principle [8].

The calculations of the modal reflectivity from the fiber end-facet is performed with a three-dimensional frequency domain modal method relying on Fourier expansion technique [9, 10]. In brief, the Fourier expansion method relies on an analytical integration of Maxwells equations along one direction (the fiber axis) and on a supercell approach in the other two transversal directions. We emphasize that for the radiated field at the surface S ₁ we have treated the elastic scattering at the end-facet fully, i.e. the incident fundamental mode is partly reflected (no mode mixing for symmetry reasons) and partly radiated/transmitted. Through the surface equivalence principle, any wave-front can be considered as a source of secondary waves which add to produce distant wave-fronts according to Huygens principle.

 

Fig. 1. Schematics of the fiber geometry and the imaginary closed surface S=S ₁+S ₂, where S ₁ is a circle parallel and close to the end-facet and S ₂ is a semi-sphere concentric with S ₁.

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In order to apply the surface equivalence principle, we surround the end-facet radiating fiber with an imaginary closed surface S=S ₁+S ₂, where S ₁ is a circle parallel and close to the end-facet and S ₂ is a semi-sphere concentric with S ₁, see Fig. 1. Both of these surfaces are of infinite extent. By considering proper surface currents on S which are equivalent in the sense that they yield the same field outside S, we can formally erase all the existing elements inside S and fill it with a perfect electric conducting (PEC) body. The equivalent surface electrical and magnetic currents on S are J e(r)=n̂×H and M_e(r)=-n̂×E, respectively, where n̂ is an outward normal to the surface S [8]. Of course, these currents radiate in the presence of the PEC and not in unbounded free-space. Because of the infinite radius of S ₂, electromagnetic fields on this surface take zero value. On S ₁ electromagnetic fields are determined from near-fields. Since the S ₁ surface is an infinitely extended flat PEC body we can utilize image theory to replace the conductor by image currents -J_e and M_e. Ultimately, computation of radiated fields of an open-ended PCF is simplified to computing radiated fields of the fictitious current source M(r)=2M_e(r)=-2n̂×E in free-space. Calculating radiated fields of the current source can be achieved systematically through electric π_e(r) and magnetic π_m(r) Hertz vector potentials defined by:

Employing these potentials, electric and magnetic fields are obtained by:

where ω=ck, k=2π/λ, and λ is the free-space wavelength. From Eqs. (2a) and (2b) we expect an improvement in computational time by a factor of roughly two compared to a direct use of the surface equivalence principle [11] in which both fictitious electric and magnetic surface currents exist.

We employ the described approach in a fully vectorial study of the evolution of the fundamental mode of a PCF from near-field to far-field. As an example we consider the fundamental mode in a pure-silica PCF with a normalized air-hole diameter d/Λ=0.45 and in our simulations we utilize the scale-invariance of Maxwell’s equations for a frequency-independent dielectric function. Recent experimental studies showed quite complicated interference phenomena in the near to far-field transition [7] and in particular a focusing behavior was observed at a finite distance from the end-facet. As illustrated in Fig. 2 this focusing phenomena is born out by our numerical simulation where the peak intensity of the radiation increases up to short distances from the fiber end-facet. The electric field intensity is maximal at a distance z0 which is typically in the range Λ≲z ₀≲10×Λ. At this particular distance the intensity pattern is rotated by an angle of π/6 compared to the near-field, in full agreement with the experimental observations. As the wavelength is increased relative to the pitch the diffraction by the six inner-most air holes increases and z0 shifts closer to the end facet of the PCF. We emphasize that this interference phenomenon has no counterparts for e.g. step-index fibers with cylindrical symmetry; the focusing relies fully on the six-fold rotational symmetry of the air-hole lattice. It should also be emphasized that the phase-front is non-trivial (non-constant) in the transverse plane at z ₀ and in that sense the focusing differs from that observed with a classical lens.

In Fig. 3 we show the electric field amplitude at various distances from the end-facet up to the asymptotic far-field region. The nearly hexagonal shape of the fundamental mode of the fiber (Panel A) is transformed into a nearly circular shape (Panel B) which is followed by a hexagonal shape at z≃z₀ (Panel C) where the orientation is rotated by an angle of π/6 with respect to the fiber mode at z=0. This hexagonal shape expands as it propagates (Panels D and E) and then becomes close to circular again above the 1/e amplitude level (Panel F). As the distance from the end-facet becomes larger, six satellites start emerging from the mode shape below the 1/e amplitude level (Panel G). Finally, for larger distances satellites appear clearly (Panel H).

 

Fig. 2. Variation of the electric field intensity with distance from the end-facet (z=0) at the center of fiber.

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As an alternative to the above rigorous approach, it is in fiber optics common to approximate the far field by a Fourier transformation of the near field [12]. Here we shall derive this expression starting from the above formalism. In the far-field limit we approximate the |r-r′| in the denominators of Eqs. (1a) and (1b) by r and in the exponential we correspondingly approximate it by r-r̂·r′, where r̂=r/r is the unit radial vector of the spherical coordinate. The magnetic Hertz vector potential, Eq. (1b), then simplifies to

where

with θ and ϕ being the polar and azimuthal angles, respectively, and r=r r̂+ϕ̂ϕ+θ$\widehat{\theta }$. The electric field now becomes

We will consider small angles θ of divergence so that r̂×M=-M_yx̂+M_xŷ+𝒪(θ) by which Eq. (5) simplifies to

with the subscript t indicating the transverse component. The integral is nothing but a twodimensional Fourier transform of the near-field corresponding to the overlap of the near field with a free-space plane-wave.

 

Fig. 3. Electric field amplitudes at distances z from the end-facet varying from z=0 (Panel A) to z=12×Λ (Panel H).

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Fig. 4 shows a plot of the asymptotic far-field intensity, including results of the rigorous surface-equivalence principle as well as the approximate far-field calculated from Eq. (6). The far-field pattern is computed for λ/Λ=0.1 at a distance of 1000×Λ from the end-facet, which is far from the starting point of the Fraunhofer region. For an extended source of electromagnetic radiation, the Fraunhofer region appears at a characteristic distance 2D ²/λ from the source where the parameter D is defined by the maximum linear dimension of the source. For a PCF D~2Λ, so Fraunhofer region sets in around 80×Λ. The inset shows a contour plot of the far-field intensity calculated with the rigorous surface-equivalence principle. We observe six low-intensity satellite peaks as well as further high-order diffraction peaks, in full agreement with experimental observations [1, 7]. As seen, the approximate results of Eq. (6) are in qualitative agreement with the full numerical solution, and it is only in the sub-1/e ² intensity regime that we observe quantitative deviations. This suggests that Eq. (6) could be a good starting point for numerical calculations of the NA and the M-squared value for LMA PCFs, which we will report elsewhere.

In conclusion we have used a surface equivalence principle to rigorously calculate radiated fields starting from fully-vectorial simulations of the near field. Our simulations have revealed a focusing behavior which is maximal at a characteristic distance, of the order 10×Λ from the end-facet. In the far-field limit we have shown how qualitative and to some degree also quantitative insight may be gained from simple two-dimensional Fourier transforms of the nearfield.

 

Fig. 4. Asymptotic far-field intensities calculated by the rigorous approach as well as the approximate expression, Eq. (6), along the high-symmetry directions ϕ=0 and ϕ=π/2. The inset shows the corresponding full contour plot obtained with the rigorous approach.

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We thank J. R. Folkenberg and P. M. W. Skovgaard (both with Crystal Fibre A/S) for stimulating discussions. N. A. M. is supported by The Danish Technical Research Council (Grant No. 26-03-0073).