Field-flattened, ring-like propagation modes

Michael J. Messerly; Paul H. Pax; Jay W. Dawson; Raymond J. Beach; John E. Heebner

doi:10.1364/OE.21.012683

1. Introduction

Most nonlinear propagation artifacts in glass waveguides can be reduced by spreading the power they carry over a large area. Many telecommunications and laser applications, however, require the power to be confined to a single transverse spatial mode. Unfortunately, as a mode’s area increases, its propagation constant approaches those of its neighboring modes, making it susceptible to power cross-coupling and potentially degrading the mode’s spatial or temporal fidelity.

Optical fibers that propagate power in a high-order mode [1,2] offer a path to simultaneously increasing the effective area [3] of a mode and the spacing between the desired mode’s propagation constant and those of its neighbors. Unfortunately, the high-order modes of a step index fiber can have hotspots – regions in their transverse profiles where the local irradiance significantly exceeds the average value – which may make them more susceptible to damage or nonlinear artifacts than modes whose power is relatively uniformly distributed, such as the fundamental.

Optical fibers having a flattened fundamental [4–8] are also attractive, as they spread the propagating power uniformly, and in an amplifier allow for uniform and efficient extraction of energy from the gain medium. Like all waveguides, though, they are bound by a mode size-spacing tradeoff, and we show below that in this regard they are only moderately better than more economically-manufactured conventional guides.

We present here a design methodology that combines the benefits of the two waveguides described above, enabling the construction of flattened high-order modes. Specifically, we provide rules for creating structures that support field-flattened segments, stitching structures that interconnect these segments, and terminating regions that bind (impedance match) the resulting engineered mode to the cladding. The effective index of these modes remains fixed as the number of rings and sizes of the rings are varied, while the effective indices of all other modes, and the group index of the flattened mode, change.

We also present examples of fibers that support field-flattened, ring-like propagation modes and compare their characteristics to the characteristics of high-order modes of step-index fibers, showing that the former are more compact and can have larger intermodal spacings. Though not discussed here, we expect that selective-doping will further discriminate the preferred flattened mode from other modes, as suggested by others for fibers having selectively-doped flattened fundamental modes [9].

2. Definitions

In the step-like structures of the following designs, the field’s continuity is enforced between steps by matching the field and its radial derivative across the interfaces. The modes of the guides are analyzed by the transfer matrices of Appendix II and by a separate two-dimensional mode solver that finds the eigenmodes of the scalar Helmholtz equation.

Most of the underlying mathematics and physics are considered in the Appendices. Appendix I presents Bessel solutions to the equation governing axially-symmetric waveguides such as a conventional telecom fiber; its results can be used to determine the refractive indices and thickness of the layers that comprise the flattened, stitching, and termination groups defined below. Appendix II presents transfer matrices that can also be used to determine layer indices and thicknesses, and to determine the properties of all bound modes of the fiber. Appendix III presents closed-form solutions to the mode normalization integral. Appendix IV defines several mode size-spacing products and shows that for a given waveguide these products are fixed, a consequence of the radiance theorem.

2.1 Scaled quantities

A characteristic numerical aperture of the fiber, NA_flat, is defined as:

N A_{f l a t} = \sqrt{n_{f l a t}^{2} - n_{c l a d}^{2}}

where n_clad is the refractive index of the cladding and n_flat is the index of the layer or layers over which the field is to be flattened. The scaled radial coordinate, v, is defined as:

v = \frac{2 π}{λ} r N A_{f l a t}

where λ is the wavelength of the guided light and r is the radial coordinate. The scaled refractive index profile, η(v), is defined as:

η (v) = [n^{2} (v) - n_{c l a d}^{2}] / N A_{f l a t}^{2}

For the flattened waveguides described here, n_flat is usually chosen to be the minimum refractive index that can be well controlled. For silica fibers, the flattened layer might be lightly doped with an index-raising dopant such as germanium or doped with a rare-earth element along with index-raising and lowering dopants. Alternatively, n_flat might be pure silica and the cladding might be lightly doped with an index depressing agent such as fluorine; in this case, the dopant only needs to extend to the penetration depth of the desired mode.

A layer group’s area-averaged index, 〈η〉, is defined as:

〈 η 〉 = \sum_{g r o u p} η_{i} A_{i} / \sum_{g r o u p} A_{i}

where η_i and A_i represent the scaled index and cross-sectional area of the i^th layer of the group. In the layer groups defined below, we sometimes constrain this value; 〈η〉 sometimes alters the number of allowed modes or the guide’s intermodal spacings.

Several of the examples that follow list a mode’s scaled effective area and illustrate its scaled field. The scaled area is defined such that the physical area, A_eff, is given by Eq. (57):

A_{e f f} = \frac{{(λ / 2 π)}^{2}}{N A_{f l a t}^{2}} A_{e f f}^{s c a l e d}

The scaled field is defined such that the physical field, ψ, is given from Eq. (50):

ψ = \frac{2 π}{λ} N A_{f l a t} P_{0}^{\frac{1}{2}} ψ_{s c a l e d}

where P₀ is the power carried by the mode.

In the following examples, η is assumed to range between ± 10, which is achievable for germanium and fluorine-doped silica provided NA_flat is on the order of 0.06. In silica, other dopants might extend this range moderately, or in phosphate glasses or holey structures, various dopants or air holes can extend this range significantly. Moreover, in holey fibers NA_flat might be controlled to a much smaller value, which would proportionally extend the range of η. A larger range of indices is generally advantageous, as it reduces the portion of the guide devoted to the stitching and matching groups described below.

2.2 Flattened layers

A flattened layer is one in which the field does not vary with radius; that is, one where:

ψ' = \partial ψ / \partial r

is zero. Equation (29) and Eq. (32) of Appendix I show that for this to occur the layer’s index must be equal to the guided mode’s effective index (n_flat = n_eff) and the azimuthal order, l, must be equal to zero. Furthermore, it is necessary that a flattened layer be joined to appropriate stitching or termination groups, as defined below.

2.3 Stitching groups

A stitching group is a layer or group of layers in which the field’s slope is zero at both endpoints (to match that of the adjacent flattened region) and is predominantly nonzero between those points, usually crossing zero one or more times. This can be accomplished in different ways to produce a variety of mode shapes; several examples are presented here.

Figure 1, Fig. 2, and Fig. 3 illustrate stitching groups that might form a portion of a guide that supports a flattened mode. In the figures, η_flat is 1 (from Eq. (3) since n(v) = n_flat), the minimum and maximum values of η are assumed to fall between ± 10, and the left edge of each group starts at v₀ = 0.5π, an arbitrarily chosen value. The thicknesses of the layers that comprise the groups were determined numerically from Bessel solutions to the wave equation, as outlined in Appendix I.

Fig. 1 Index profiles of portions of a flattened waveguide, illustrating half-wave stitching. The black curves represent scaled index, the grey curves represent the field of the flattened mode (arbitrary units), and the tables list the scaled thickness and index of each group’s layers. The left edge of the first layer of each group is placed at v = 0.5π and on both sides of all groups η = 1. On the left sides of all groups ψ = 1 and ψ´ = 0; on the right side of (a), ψ = −0.78 and ψ´ = 0, and on the right sides of (b) and (c), ψ = −1 and ψ´ = 0. All quantities are dimensionless.

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Fig. 2 Index profiles of portions of a flattened waveguide, illustrating full-wave stitching. The black curves represent scaled index, the grey curves represent the corresponding field of the flattened mode (arbitrary units), and the tables list the scaled thickness and index of each group’s layers. The left edge of the first layer of each group is placed at v = 0.5π and on both sides of all groups η = 1. On the left sides of all groups ψ = 1 and ψ´ = 0; on the right side of (a), ψ = 0.66 and ψ´ = 0, and on the right sides of (b) and (c), ψ = 1 and ψ´ = 0. The minimum fields for examples (a), (b) and (c) are −0.78, −0.78, and −0.71, respectively. All quantities are dimensionless.

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Fig. 3 Index profiles of portions of a flattened waveguide, illustrating fractional wave stitching. The black curves represent scaled index, the grey curves represent the field of the flattened mode (arbitrary units), and the tables list the scaled thickness and index of each group’s layers. The left edge of the first layer of each group is placed at v = 0.5π and on the right side of a) and both sides of (b) and (c), η = 1. On the left sides of (b) and (c), ψ = 1 and ψ´ = 0; on the right side of all groups, ψ = 1 and ψ´ = 0. All quantities are dimensionless.

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2.3.1 Half-wave stitching

Figure 1 illustrates three half-wave stitching groups, that is, three groups that cause the field’s polarity to change sign an odd number of times.

Figure 1(a) shows half-wave stitching accomplished with a single layer. The field changes by a factor of −0.78 as determined by its Bessel solution’s behavior. Simulations show that for a single layer, as the left side of the group is placed at higher values of v₀, the ratio of the magnitude of the fields approaches unity and:

\lim_{v_{0} \to \infty} (Δ v \sqrt{η - 1}) = m π

where Δv is the scaled thickness of the layer, η is the layer’s scaled index, the numeral one arises from the assumption that the layer is surrounded by field-flattened layers having η = 1, and m is an odd integer. This is also the condition for single layer, half-wave stitching in a one-dimensional slab waveguide (in that case, independent of v₀).

In Fig. 1(b) a second layer is added to make the magnitude of the field to the right of the group the same as the magnitude to its left. We mention without illustration that if the sequence of the layers in Fig. 1(b) is reversed – that is, if the higher index layer is place to the right of the lower index layer – the field on that group’s right can be made an even smaller fraction of the field on its left, when compared to the single layer example of Fig. 1(a).

Figure 1(c) illustrates an evanescent half-wave stitching group, a term that here refers to groups having at least one layer in which the field is the sum of exponentially growing and decaying functions. The thicknesses of the layers that comprise the group are adjusted to also make the 〈η〉 = 1 for the group (see Eq. (4)) and to make ψ = −1 and ψ´ = 0 on the group’s right edge.

2.3.2 Full-wave stitching

Figure 2 illustrates three full-wave stitching groups, that is, three groups that cause the field’s polarity to change sign an even number of times.

Figure 2(a) shows half-wave stitching accomplished with a single layer. The field changes by a factor of 0.66 due to its Bessel solution’s behavior. As v₀ is increased an equation similar to Eq. (8) holds, but whose right-hand side is proportional to an even multiple of π.

Figure 2(b) illustrates a two layer full-wave group that returns the field’s magnitude and polarity to their original values. The thickness of the group’s first layer is chosen to make the field zero at the right boundary of the first layer. The thickness and index of the second layer are determined numerically to make ψ = 1 and ψ´ = 0 on the group’s right edge.

Figure 2(c) illustrates a five-layer evanescent full-wave stitching group. The thickness of the first two layers and a portion of the thickness of the third layer are chosen so that the slope is returned to zero, the field is changed by a factor of −0.707 (ψ² drops by a factor of two) within the third layer; we also require that, for the group, 〈η〉 = 1 (see Eq. (4)). The thicknesses of the second portion of the third layer and of the remaining two layers are determined in the same fashion, but now with the constraint that ψ = 1 and ψ´ = 0 on the group’s right edge.

2.3.3 Fractional-wave stitching

Figure 3 illustrates three fractional wave stitching groups, that is, three groups that return the field’s slope to zero without allowing the field’s polarity change to sign.

Figure 3(a) illustrates a central stitching layer. The central index is lower than the cladding’s and the field consequently grows exponentially with position; the field on-axis is not zero, here it is 2% of the field at the layer’s edge, and hence it is not classified as a half-wave group. Simulations show that layers such as this can efficiently disrupt the properties of a guide’s non-flattened mode, though their disadvantage is that they carry very little power. Note that the central index of Fig. 3(a) could be made higher than the cladding’s index, resulting in a field similar to that in Fig. 1(a) or Fig. 2(a).

Figure 3(b) illustrates a three layer stitching group in which the field dips but does not pass through zero, creating an effect similar to those created by other structures such as those suggested in [10] (though the index dips of those structures do not return the field or slope to their original values). Simulations suggest that groups such as those in Fig. 3(b) may be difficult to manufacture since their behavior varies strongly with the layers’ thicknesses.

Figure 3(c) illustrates a three layer stitching group in which the field’s magnitude rises within the group. The resulting hotspot may be advantageous for applications where field effects are to be enhanced, but problematic for most other high power laser applications. Like the example of Fig. 3(b), simulations suggest that such a group may be difficult to manufacture.

2.4 Termination groups

A termination group is a layer or group of layers placed between one region of a guide, here most often a region in which the slope of the desired mode’s field is zero, and the guide’s cladding. The indices and thicknesses of the layers that comprise the group are chosen to force the cladding’s exponentially-growing term to zero, and to thus bind the mode to the guide. Termination (binding) is analogous to impedance matching.

The examples of this and the following section give the flattened mode’s scaled effective area and illustrate its scaled field, quantities defined by Eq. (5) and Eq. (6). For example and comparison, consider a step-index fiber that supports the LP₀₁ mode and is at the cusp of supporting the LP₀₂ mode, that is, v = 1.23π. It can be shown that its fundamental mode has a scaled effective area of 37.5; therefore, if the guide’s design operates at λ = 1 μm and its core has a numerical aperture of 0.06, its effective area will be 260 μm². It can also be shown that this mode has a scaled peak field of 0.219 = 1/√20.8. If the fiber carries 1 kW of power its peak field will be 2.61 W^1/2/μm and its peak irradiance will be (2.61 W^1/2/μm)² = 6.8 W/μm². Note that the peak irradiance is 1.8 times higher than the simple ratio of the power to the effective area (37.5 ÷ 20.8). For flattened modes, this ratio is closer to unity, typically 1.15.

Figure 4 illustrates three termination groups applied to three flattened waveguides. In the figure, η_flat is 1 (from Eq. (3) since n(v) = n_flat) and the minimum and maximum values of η are limited to ± 10. The thickness of the flattened layer is chosen so that each guide is on the cusp of allowing one axially-symmetric mode beyond the flattened mode. The thicknesses of the layers that comprise the groups were determined numerically from Bessel solutions to the wave equation, applying the constraints listed for each example.

Fig. 4 Termination layers applied to three simple flattened modes. The black curves represent scaled index and the grey curves represent the scaled field. In (a) the scaled area is 47 (converted to effective area via Eq. (5)) and the scaled peak field is 1/√41 (converted to field via Eq. (6)); in (b) the scaled area is 46 and the scaled peak field is 1/√42; and in (c) the scaled area is 53 and the scaled peak field is 1/√46. All quantities are dimensionless.

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Figure 4(a) illustrates a single-layer termination group. Note that the field extends relatively far into the cladding; at the cladding interface the field is 93% of its value in the flattened region and 21% of the mode’s power is guided in the cladding.

Since the effective index of the guide’s flattened mode is predetermined (because n_eff = n_flat), the mode’s decay constant in the cladding is fixed and consequently the field in the cladding can only be reduced by reducing the field at the cladding interface – the purpose of the additional layers in Fig. 4(b) and Fig. 4(c).

Figure 4(b) illustrates a two-layer termination group, similar to those described in [8]. In this group, the group-averaged scaled index, Eq. (3), serves as an additional constraint; simulations show that it strongly affects the field at the cladding interface. In the example, the layers’ thicknesses are varied to make the field at the cladding boundary 50% of the field in the flattened layer (this occurs with the group’s average index, Eq. (4), set to 〈η〉 = 0.7), and to match the field’s slope at the cladding interface. Roughly 7% of the mode’s power is guided in the cladding.

Figure 4(c) illustrates a three-layer termination group. The field is set to zero at the interface between the first and second layer, the local minima in the second layer is 50% of the field in the flattened layer, and the group-averaged index, Eq. (4), is set to 〈η〉 = 0.7. The field at the cladding interface is −3% of the field in the flattened region, and 0.04% of the mode’s power is guided in the cladding, though now a significant power-fraction is guided by the termination group.

Termination groups of the type shown in Fig. 4(c) enhance the mode’s confinement but also allow at least one additional axially-symmetric mode, plus the asymmetric modes that may accompany it. Relative to the desired mode, the additional modes can have very different propagation constants, very different transverse power distributions, or both; thus, they may not readily couple to the desired mode and may not be problematic.

3. Example waveguides

Waveguides that propagate a flattened high order mode are created by interleaving flattening layers with stitching groups, typically starting from the inside of the guide and working outward, then binding the mode to the cladding with a termination group.

Table 1 lists designs for three waveguides; A and B both support a three-ringed, flattened mode, and C supports several higher-order modes. A and B each have three flattened layers (i, v and ix), two three-layer half-wave stitching groups similar to those illustrated in Fig. 1(c) (ii-iv and vi-viii), and a two-layer termination group similar to the one in Fig. 4(b) (x-xi). Surrounding these layers is the cladding having η = 0.

Table 1. Parameters for two three-ringed flattened mode designs (A and B) and a step-index design (C). All quantities are dimensionless.

View Table

In Design A the flattened layers have equal cross-sectional areas, both stitching groups have 〈η〉 = 3.0, and the termination group has 〈η〉 = 0.7. In Design B the flattened layers have equal widths, both stitching groups have 〈η〉 = 2.4, and the termination group has 〈η〉 = 0.7.

We compare the flattened LP₀₃ modes of Designs A and B to the LP₀₃ mode of a few-mode step index design, Design C. Design C is similar to the high-order mode fibers reported by others [2], but has a smaller v-number to make its mode count similar to those of A and B.

Figure 5 shows line-outs of the field distributions for the three designs. For Design A, the scaled area is 140 and the scaled peak field is 1/√122; for Design B the values are 150 and 1/√134; and for Design C the values are 140 and 1/√30.8. Thus for equivalent areas, the peak intensity of the modes in Designs A and B would be one-fourth the peak intensity of the mode of Design C, greatly reducing the damage threshold. (The large disparity between the two measures of mode size for C – 140 for its effective area vs. 30.8 for the reciprocal of its peak irradiance, a ratio of 4.5 – is due to its central hotspot.)

Fig. 5 Line-outs of the scaled index (dark lines) and field (grey lines) for the three designs; a), b), and c) correspond to Designs A, B, and C. All quantities are dimensionless.

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Figure 6 shows the transverse field distributions of the LP₀₃ and LP₁₃ modes of the three designs; when bent, the LP₀₃’s will morph toward their respective LP₁₃’s. Note that the power is more compactly packed in the flattened modes than in the step-index mode.

Fig. 6 Field (not irradiance) distributions for the LP03 and LP13 modes of the three example designs – two flattened-mode fibers and a step index fiber. The colors blue and red designate positive and negative polarities of the field and the depth of the color designates its relative amplitude. All figures are scaled as the one on the left, and all quantities are dimensionless.

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Note, too, that the inner rings of LP₁₃ modes of the flattened designs have essentially the same diameter as the inner rings of their corresponding LP₀₃ modes. The inner ring of the LP₁₃ mode for the step-index design, though, has a substantially larger diameter than its corresponding LP₀₃ mode. This suggests the latter’s mode will experience a larger shift in its centroid when that fiber is bent. The design of the high-order mode fiber in [2] has a central spike in its index profile, perhaps to keep its mode centered.

Figure 7 compares the size-spacing products, Θ_eff (the phase index-area spacing – essentially the radiance), defined by Eq. (59) in Appendix IV, for the modes of the three designs. The size-spacing products are an invariant of a design. Larger values are likely preferable, since they imply that larger-sized modes may be fabricated while keeping the intermodal spacing constant, and thus keeping the likelihood of intermodal coupling constant. Bear in mind that the effective area term in the Θ_eff equation is the same for all of a design’s modes; for each design here, it is chosen to be the area of the design’s LP₀₃ mode.

Fig. 7 Plots (a) through (c) show, as a function of the azimuthal order l, the size-spacing products for the effective indices of the modes of the three designs (Θ_eff is defined in Eq. (59)). The red circles designate the LP₀₃ mode, which for A and B is the flattened mode and the red arrow in (b) highlights the relatively large spacing that has been created between the LP₁₂ and LP₂₂ modes of design B. For all of a design’s modes, the value of A_eff used to calculate its size-spacing products is the area of that design’s LP₀₃ mode. The legend adjacent to (c) applies to all figures, and all quantities are dimensionless.

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For Designs A and B, the spacing between the Θ_eff’s for the three highest-order symmetric modes, the LP₀₂, LP₀₃ (flattened mode) and LP₀₄ (on the cusp of existence), have been made equal by choosing an appropriate thickness for the flattened layers and by choosing an appropriate value of 〈η〉 (Eq. (3)) for each design’s stitching groups.

For A and B, the size-spacing differential for the axially-symmetric modes is 2.5 times larger than it is for Design C, and three times larger than for the designs in Fig. 4. This implies that for the same manufacturing tolerances, the three-ringed flattened design can have 2.5 times the area of C, or three times the area of the designs in Fig. 4.

Note that the effective index spectra of A and B are strongly affected by the relative widths of the flattened layers; a relatively large spacing has been created between the LP₁₂ and LP₂₂ modes of B (red arrow in Fig. 7(b)).

Figure 8 compares the size-spacing products, Θ_g (group index-area spacing), defined by Eq. (62) in Appendix IV, for the modes of the three designs. The size-spacing products are an invariant of a design. Larger values are likely preferable, since they imply that larger-sized modes may be fabricated while the keeping the intermodal spacing constant, and thus keeping the likelihood of intermodal coupling constant. Keep in mind that the effective area term in Θ_g equation is the same for all of a design’s modes; for each design, it is chosen to be the area of the design’s LP₀₃ mode.

Fig. 8 Plots (a) through (c) show, as a function of the azimuthal order l, the size-spacing products for the effective indices of the modes of the three designs (Θ_g is defined in Eq. (62)). The red circles designate the flattened mode, which for A and B is the flattened mode, and the red arrow in (b) highlights the significant increase in the spacing between the group velocities of the LP₁₂ and LP₂₂ modes by design B. For all of a design’s modes, the value of A_eff used to calculate its size-spacing products is the area of that design’s LP₀₃ mode. The legend adjacent to (c) applies to all figures, and all quantities are dimensionless.

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Note that the group index spacings of the two flattened designs, A and B, are significantly larger than those of the step-index design, C; the larger spacings may help reduce linear and nonlinear modal coupling in pulsed laser applications. Simulations show that the group delay spectra of A and B are strongly affected by the relative widths of the flattened layers. Note that a local maxima has been created for the LP₁₂ mode of B (red arrow in Fig. 8(b)), and that in A and C the flattened mode is the slowest axially-symmetric mode, while in B it is the fastest of all modes.

4. Discussion

The design philosophy presented here is atypical – it begins with the desired mode’s shape and then constructs a waveguide that will allow it. Flattening layers are interleaved with stitching groups and a termination group binds the flattened mode to the guide; the latter is analogous to impedance matching. For axially-symmetric waveguides, the thicknesses or indices of the layers that comprise the stitching groups must be changed when the group’s radial placement is changed; the examples presented here should be considered starting points for user-specific designs.

The high-order flattened modes allow two size-spacing invariants – one relating to the phase index spacing, one relating to the group index spacing – to be tailored. In particular, we have shown that the effective index (phase index) spacing of the guide’s axially-symmetric modes can be increased substantially, and show that this spacing grows in proportion to the number of rings added to the structure.

Note that the flattened modes do not suffer potentially problematic hotspots, they inherently pack the propagated power into a compact cross-section, and they may reduce a mode’s susceptibility to some artifacts such as nonlinear self-focusing. In an amplifier, they allow power to be extracted uniformly and efficiently across the mode’s cross section. Furthermore, in amplifier applications the stitching and termination groups would not likely be doped with rare-earth ions, allowing for better control of their indices, and since the field of the flattened mode is near-zero in those regions, avoiding leaving regions of unsaturated gain that might contribute to noise or amplification of undesired modes.

Here we have qualitatively considered the bending properties of the flattened high-order modes by inspecting the transverse structure of the neighboring mode that they would couple to, and find that the flattened modes will stay well-centered. Quantitative bend-loss studies are in progress.

Comparisons to the high-order modes of a step-index fiber are complicated by the fact that the effective area, as conventionally defined, does not account for hotspots in a mode’s peak irradiance. We have used the effective area metric here though in some applications it may give an overly optimistic representation of the performance of high order step-index modes. Despite this (sometimes) lenient metric, the high-order mode of the step-index example fiber is less attractive than the flattened modes described here in terms of intermodal spacing, peak irradiance, and the compactness. While increasing the v-number of the step-index design would improve the intermodal spacing, it would also increase its mode count, accentuate its central hotspot, and further reduce its mode’s packing density.

In principle, flattened high-order modes could be manufactured with conventional telecom techniques such as modified chemical vapor deposition and outside vapor deposition, but the tighter manufacturing tolerances allowed by holey-fiber construction techniques may, however, be preferable or necessary.

Appendix I: Bessel solutions

Consider the equation that governs the radially-varying portion of the field in an axially symmetric waveguide such as a conventional telecom optical fiber [11]:

{\frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial}{\partial r} - \frac{l^{2}}{r^{2}} + {(\frac{2 π}{λ})}^{2} [n^{2} (r) - n_{e f f}^{2}]} ψ (r) = 0

where ψ represents the field of a guided mode, l is the azimuthal order, n(r) is the index at radial coordinate r, n_eff is the effective index (propagation constant) of the mode, and λ is the vacuum wavelength of the guided light. In the discussion that follows, we assume that the radial index profile varies in discreet steps, or layers.

Define the dimensionless and scaled variables:

v = \frac{2 π}{λ} r \sqrt{n_{f l a t}^{2} - n_{c l a d}^{2}}

η = \frac{n^{2} (v) - n_{c l a d}^{2}}{n_{f l a t}^{2} - n_{c l a d}^{2}}

and:

η_{e f f} = \frac{n_{e f f}^{2} - n_{c l a d}^{2}}{n_{f l a t}^{2} - n_{c l a d}^{2}}

where n_flat is the refractive index of the layer or layers in which the field will ultimately be flattened (in the method prescribed in this paper, n_flat is chosen before the waveguide is designed). In these terms, the wave equation becomes:

{\frac{\partial^{2}}{\partial v^{2}} + \frac{1}{v} \frac{\partial}{\partial v} - \frac{l^{2}}{v^{2}} + η (v) - η_{e f f}} ψ (v) = 0

For weak waveguides, the field and its radial derivative are continuous across the step-like boundaries between layers. Since the radial derivative is continuous, so is the quantity:

ζ = r \frac{\partial ψ}{\partial r} = v \frac{\partial ψ}{\partial v}

To determine the field distribution of the modes of a complex waveguide, we track ψ and ζ; we begin by determining analytic solutions for the field in layers whose index is greater than, less than, and equal to the propagation constant. Each analytic solution has two unknown constants, which can be determined by the boundary conditions.

Begin by considering layers that are neither the inner-most layer, here referred to as the “core,” nor the outermost layer, referred to as the “cladding.” The cladding is presumed to extend to infinity.

In layers where η > η_eff (n > n_eff), the solution to the wave equation is:

ψ (x) = A J_{l} (x) + B Y_{l} (x) (n > n_{e f f})

where J_l and Y_l are oscillatory Bessel functions, A and B are unknown constants, and:

x = v \sqrt{| η - η_{e f f} |}

If ψ and ζ are known at some position v₁, such as at one of the layer’s boundaries, then A and B can be expressed:

A = \frac{π}{2} [x_{1} Y_{l}' (x_{1}) ψ_{1} - Y_{l} (x_{1}) ζ_{1}]

B = \frac{π}{2} [- x_{1} J_{l}' (x_{1}) ψ_{1} + J_{l} (x_{1}) ζ_{1}]

A and B were determined with the help of the identity [12]:

J_{l} (x) \cdot x Y_{l}' (x) - x J_{l}' (x) \cdot Y_{l} (x) = 2 / π

Note that the derivatives of the Bessel functions can calculated exactly from the identities:

x J_{l}' (x) = l J_{l} (x) - x J_{l + 1} (x)

x Y_{l}' (x) = l Y_{l} (x) - x Y_{l + 1} (x)

In layers where η < η_eff (n < n_eff) the solution to the wave equation is:

ψ (x) = A I_{l} (x) + B K_{l} (x) (n < n_{e f f})

where I_l and K_l are exponentially growing and decaying modified Bessel functions and A and B are unknown constants. If ψ and ζ are known at some position v₁, such as at one of the layer’s boundaries, then A and B can be expressed:

A = - x_{1} K_{l}' (x_{1}) ψ_{1} + K_{l} (x_{1}) ζ_{1}

B = x_{1} I_{l}' (x_{1}) ψ_{1} - I_{l} (x_{1}) ζ_{1}

In determining A and B we used the identity:

K_{l} (x) \cdot x I_{l}' (x) - x K_{l}' (x) \cdot I_{l} (x) = 1

Note that the derivatives of the Bessel functions can be calculated exactly from the identities:

x I_{l}' (x) = l I_{l} (x) + x I_{l + 1} (x)

x K_{l}' (x) = l K_{l} (x) - x K_{l + 1} (x)

In layers where η = η_eff (n = n_eff) the wave equation reduces to:

{\frac{\partial^{2}}{\partial v^{2}} + \frac{1}{v} \frac{\partial}{\partial v} - \frac{l^{2}}{v^{2}}} ψ (v) = 0

For l ≠ 0 the solution is:

​ ψ = A v^{+ l} + B v^{- l} ​ ​ ​ ​ (n = n_{e f f}, l \neq 0)

and the constants A and B become:

A = \frac{v_{1}^{- l}}{2 l} (l ψ_{1} + ζ_{1})

B = \frac{v_{1}^{l}}{2 l} (l ψ_{1} - ζ_{1})

For l = 0 the solution is:

​ ψ = A + B \ln (v) ​ (n = n_{e f f}, l = 0)

and the constants A and B become:

A = ψ_{1} - ζ_{1} \ln (v_{1})

B = ζ_{1}

Note that in Eq. (32), the field can be made independent of position by forcing the constant B to zero (from Eq. (34), this is equivalent to making the field’s slope zero); thus a necessary condition is that n = n_eff. Comparing Eq. (29) and Eq. (32) we see that the field can only be flattened if, in addition to n = n_eff, the azimuthal order, l, is also zero.

Now consider the inner-most layer, the core, and the outer-most layer, the cladding. In these, only a single Bessel solution is allowed. In the core the solutions are:

ψ = A J_{l} (x) (η > η_{e f f})

ψ = A I_{l} (x) (η < η_{e f f})

ψ = A v^{l} (η = η_{e f f}, l \neq 0)

ψ = A (η = η_{e f f}, l = 0)

and in the cladding the allowed solution is:

ψ = A K_{l} (x)

Appendix II: Transfer matrices

The solutions for the constants A and B can be substituted into the original expressions for ψ and the corresponding expressions for ζ to obtain transfer matrices, M, that relate ψ and ζ at position v₂ to their known values at position v₁:

[\begin{matrix} ψ_{2} \\ ζ_{2} \end{matrix}] = M [\begin{matrix} ψ_{1} \\ ζ_{1} \end{matrix}]

In all cases, the matrices can be written in the form:

M = m^{- 1} (x_{2}) m (x_{1})

where x₁ is the quantity x, defined by Eq. (16), evaluated at position v₁ and index η₁₂ (the index between v₁ and v₂), and x₂ is x evaluated at v₂ and index η₁₂.

The determinant of each matrix is unity, but they are not orthogonal. Their inverses are found by exchanging their diagonal elements and changing the signs of their off-diagonal elements.

In layers where η > η_eff (n > n_eff):

m (x) = \sqrt{\frac{π}{2}} [\begin{matrix} x Y_{l}' (x) & - Y_{l} (x) \\ - x J_{l}' (x) & J_{l} (x) \end{matrix}]

In layers where η < η_eff:

m (x) = [\begin{matrix} x I_{l}' (x) & - I_{l} (x) \\ - x K_{l}' (x) & K_{l} (x) \end{matrix}]

In layers where η = η_eff and l ≠ 0:

m (x) = \frac{1}{\sqrt{2}} [\begin{matrix} v^{- l} & 1 / l v^{l} \\ - l v^{l} & v^{l} \end{matrix}]

In layers where η = η_eff and l = 0:

m (x) = [\begin{matrix} 1 & - \ln (v) \\ 0 & 1 \end{matrix}]

The transfer matrix solution to the wave equation for a step-like fiber then becomes:

M [\begin{matrix} 1 \\ Ω_{c o r e} \end{matrix}] = (c o n s t) [\begin{matrix} 1 \\ Ω_{c l a d} \end{matrix}]

where the quantity Ω is defined as:

Ω = ζ / ψ

and Ω_clad is (from Eq. (39)):

Ω_{c l a d} = {\frac{x K_{l}' (x)}{K_{l} (x)} |}_{x = x_{c l a d}}

where x_clad is the term x, as defined by Eq. (16), evaluated at position v_clad and index η_clad = 0. Note that the Bessel derivates can be calculated from Eq. (27). Ω_core is similarly calculated from Eq. (35), Eq. (36), Eq. (37), or Eq. (38) at the core’s boundary.

The matrix M is the product of the matrices that represent the layers between the core and cladding; it takes advantage of the fact that ψ and ζ are continuous across layer boundaries. For a given waveguide, the propagation constant η_eff is determined iteratively – that is, by varying its value until the transfer matrix solution is satisfied.

In the above, (const) refers to a multiplicative constant related to the total power carried by a mode, as discussed in the following Appendix.

Appendix III: Mode normalization

This appendix gives closed-form solutions for the mode normalization integral, and defines scaled fields.

Mode normalization involves choosing the (const) term of Eq. (46) to make the power carried by a mode equal to some preselected value, P₀:

2 π {(c o n s t)}^{2} \int_{0}^{\infty} ψ^{2} r d r = P_{0}

Define ψ_scaled such that:

ψ^{2} = {(\frac{2 π}{λ})}^{2} (n_{f l a t}^{2} - n_{c l a d}^{2}) P_{0} ψ_{s c a l e d}^{2}

Then normalization reduces to setting:

2 π {(c o n s t)}^{2} \int_{0}^{\infty} ψ_{s c a l e d}^{2} v d v = 1

The integration is typically performed numerically, though with the expressions that follow, which we believe are novel, it can be calculated analytically. The solutions were obtained by integrating the above expression by parts twice and taking advantage of the fact that the bound modes’ fields satisfy the original wave equation, Eq. (13).

For η ≠ η_eff (n ≠ n_eff):

2 π \int ψ^{2} v d v = π \frac{ζ^{2} - l^{2} ψ^{2}}{η - η_{e f f}} + π v^{2} ψ^{2}

For η = η_eff and l = 0:

2 π \int ψ^{2} v d v = \frac{π v^{2}}{2} (ζ^{2} - 2 ψ ζ) + π v^{2} ψ^{2}

For η = η_eff and l = 1:

2 π \int ψ^{2} v d v = π {(\frac{v}{2})}^{2} [\frac{1}{2} {(ζ + ψ)}^{2} + 2 {(ζ - ψ)}^{2} \ln (v) - 2 (ζ^{2} + ψ^{2})] + π v^{2} ψ^{2}

And finally, for η = η_eff and l ≥ 2:

2 π \int ψ^{2} v d v = π {(\frac{v}{2 l})}^{2} [\frac{1}{l + 1} {(ζ + l ψ)}^{2} - \frac{1}{l - 1} {(ζ - l ψ)}^{2} - 2 (ζ^{2} + l^{2} ψ^{2})] + π v^{2} ψ^{2}

These are the indefinite solutions to the integrals; the contribution from an individual layer is found by evaluating its solution (depending on its index relative to the propagation constant) at the its boundaries, and subtracting one from the other. The full integral (from zero to infinity) is found by summing the individual contributions. Note that, for any waveguide design, the right-most terms of the piece-wise integrals contributes the following series to the full integral:

π {[v^{2} ψ^{2}]}_{0}^{v_{1}} + π {[v^{2} ψ^{2}]}_{v_{1}}^{v_{2}} + ... + π {[v^{2} ψ^{2}]}_{v_{c l a d}}^{\infty}

However, since v and ψ are continuous across interfaces, this reduces to

π {[v^{2} ψ^{2}]}_{0}^{\infty}

, which is zero for all bound modes. Thus, while the right-most terms contribute to the piece-wise integrals, they do not contribute to the full integral.

The closed form solutions can also be used to quickly calculate the group index of a mode via Eq. (60).

Appendix IV: Size-spacing products

This appendix defines several mode size-spacing products and shows that for a given waveguide design, these are fixed. It refers to scaled terms defined in Appendix I.

Once the scaled index profile (Eq. (11)) is specified, the scaled propagation constants, Eq. (12), and the shapes of the allowed modes are completely determined, as implied by the form of the scaled wave equation, Eq. (13). To relate scaled quantities to those that can be measured in a laboratory, begin by noting that the effective mode area can be written:

A_{e f f} = 2 π \frac{(\int ψ^{2} r d r)^{2}}{\int ψ^{4} r d r} = \frac{{(λ / 2 π)}^{2}}{n_{f l a t}^{2} - n_{c l a d}^{2}} A_{e f f}^{s c a l e d}

where the scaled effective area is defined as:

A_{e f f}^{s c a l e d} = 2 π \frac{(\int ψ^{2} v d v)^{2}}{\int ψ^{4} v d v}

For each allowed mode of a design, the propagation constant and scaled area are fixed, and thus their product, represented here by the symbol Θ_eff, is also fixed:

Θ_{e f f} = η_{e f f} A_{e f f}^{s c a l e d} = \frac{A_{e f f}}{λ^{2}} (n_{e f f}^{2} - n_{c l a d}^{2})

The right-most term is found through substitution; note that though it was derived from scaling arguments, it consists only of quantities that can be directly measured, and that since Θ_eff is fixed, if a mode’s size is increased, its effective index necessarily approaches the cladding index. Since this holds for all modes, it follows that as a desired mode’s size is increased, the effective indices of all modes necessarily approach each other.

The effective index is the phase index of the mode. When evaluating pulse propagation effects, the group index, n_g, is also important. Using an integral form of the group index [13] it can be shown that:

\frac{n_{e f f} n_{g} - n_{c l a d}^{2}}{n_{f l a t}^{2} - n_{c l a d}^{2}} = \frac{\int η ψ^{2} v d v}{\int ψ^{2} v d v}

and following arguments similar to those that led to Θ_eff, it can be shown that the following quantity is also fixed for each mode of a waveguide:

Θ_{e f f, g} = \frac{A_{e f f}}{λ^{2}} (n_{e f f} n_{g} - n_{c l a d}^{2})

where n_effn_g is the product of a mode’s phase and group indices. Like Θ_eff,, this is a strict invariant of a design (within the strictures of the weak-guiding approximation), but unfortunately the separations between the Θ_eff,g’s are not obvious indicators of the separations between the group indices. The following term is more transparent:

Θ_{g} = \frac{A_{e f f}}{λ^{2}} (n_{g}^{2} - n_{c l a d}^{2}) = 2 Θ_{e f f, g} - Θ_{e f f} + \frac{λ^{2}}{n_{e f f}^{2} A_{e f f}} (Θ_{e f f, g} - Θ_{e f f})^{2}

where the right hand side has been found by substitution. Since Θ_g depends on A_eff it is not a true invariant of the guide. However, if the mode’s area is sufficiently large (usually the case for high power laser applications) then the third term can be neglected and since Θ_eff and Θ_eff,_g are true invariants, then Θ_g is approximately invariant.

Acknowledgments

This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

References and links

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	Design A		Design B		Design C
layer	Δv/π	η	Δv/π	η	Δv/π	η
i	0.900	1	0.470	1	3.240	1
ii	0.128	10	0.133	10
iii	0.124	−10	0.137	−10
iv	0.107	10	0.099	10
v	0.289	1	0.470	1
vi	0.123	10	0.120	10
vii	0.125	−10	0.138	−10
viii	0.110	10	0.106	10
ix	0.202	1	0.470	1
x	0.076	10	0.076	10
xi	0.064	−10	0.064	−10

Field-flattened, ring-like propagation modes

Abstract

1. Introduction

2. Definitions

2.1 Scaled quantities

2.2 Flattened layers

2.3 Stitching groups

2.3.1 Half-wave stitching

2.3.2 Full-wave stitching

2.3.3 Fractional-wave stitching

2.4 Termination groups

3. Example waveguides

4. Discussion

Appendix I: Bessel solutions

Appendix II: Transfer matrices

Appendix III: Mode normalization

Appendix IV: Size-spacing products

Acknowledgments

References and links

Cited By

Figures (8)

Tables (1)

Equations (62)

Optics Express