## Abstract

It is well known that a fiber taper—an optical fiber whose size varies along its length—must exceed a certain minimum length to be adiabatic and low-loss. In contrast, we show that an optical fiber with a logarithmic refractive index profile can be adiabatically tapered over any length, however short. Its mode field distribution is independent of the fiber’s size and so remains the same along a taper. We report an experimental fiber in which tapers shorter than 2 mm can have losses as low as 0.03 dB. The fiber is compatible with standard telecoms fiber but with low bend loss and should have applications where it is desirable to make tapered fiber components (such as fused couplers and photonic lanterns) as short as possible. The index profile is analogous to the logarithmic potential of quantum mechanics.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

## 1. INTRODUCTION

A fiber taper is a piece of optical fiber whose transverse scale changes along its length. Typically, it is formed by locally heating a fiber above the softening temperature of the glass while simultaneously stretching the fiber so that the softened material narrows down. The result is a structure with a narrow uniform waist region connected to untapered pieces of fiber by two taper transitions [1,2]. Fiber tapers can be used to transform the size and/or shape of the mode pattern in a fiber [3,4]; to split, combine, or mix light waves in multiple fibers [5–8]; or to controllably couple light into structures such as optical resonators [9,10].

For many of these applications, it is important that light entering the taper in the fiber’s fundamental mode remains in the local fundamental mode all along the taper, even though the waveguide itself may change profoundly. In a single-mode fiber, any light that is coupled to higher-order modes in the taper will not be guided in the core of the output fiber, and so will be lost. Such local-mode coupling can be avoided if the taper transition is long enough; light in the fundamental mode then remains in that mode throughout, and the taper is described as being adiabatic [11,12]. This limitation on the shortness of tapers constrains the minimum size of components such as fused couplers [5] and limits the scalability of spatial multiplexers such as mode-selective photonic lanterns [6–8].

Here we report a fiber with a logarithmic refractive index profile in which the minimum length of an adiabatic taper is zero. That is, any axi-symmetric taper in the fiber is adiabatic, however short. Although the fiber design is idealized and cannot be made in practice, we have made an experimental version of the fiber that, despite practical compromises, yields a low-loss taper with a length and profile that cause high loss in standard telecoms fiber. We reported preliminary results on these fibers at the postdeadline session of OFC 2017 [13].

## 2. THEORY

#### A. Why Can’t All Tapers Be Adiabatic?

Mathematical criteria exist that delimit the local taper angle in adiabatic tapers. The smaller this angle is, the longer the taper transition must be to accomplish a given change in the size of the fiber. The most useful criterion for us is derived from the “weak power transfer” criterion of Love *et al.* [11] with our own modifications [6],

Note how the allowable size of the taper angle or gradient $d\rho /dz$ in an adiabatic taper is restricted by $\partial \mathrm{\Psi}/\partial \rho $, the rate of change of the mode field distribution with the scale of the fiber. Hence, the reason why adiabatic tapers must be longer than a minimum length is because the mode field distribution changes along the fiber [6].

As an example of a standard single-mode fiber in widespread use for telecoms applications (and also because we used this fiber in our experiments) we modelled the behavior of the fiber Corning SMF-28 for the wavelength $\lambda =1550\text{\hspace{0.17em}}\mathrm{nm}$. We assumed it to have a step-index profile, with core and cladding diameters of 8.2 and 125 μm, respectively, and an index step of $4.8\times {10}^{-3}$. These values were based on the manufacturer’s specifications [14], although the index step was deduced from the specified cutoff wavelength rather than the specified numerical aperture. They are consistent with the specified mode field diameter (MFD) of 10.4 μm at 1550 nm. We will call this the “step-profile fiber,” as its qualitative behavior is typical, though of course other choices of parameters lead to different quantitative results.

For this model of step-profile fiber, Fig. 1(a) depicts the calculated fundamental-mode field distribution along a taper. The diameter dependence of its Petermann-II MFD [15] is plotted in Fig. 2. Both the size and shape of the mode field vary substantially with fiber diameter, limiting how short an adiabatic taper can be. In fact, the curve is the same for any step-profile fiber with the same ratio of cladding and core diameters, with the axes scaled according to the V parameter [12]. Shrinking the cladding [6], grading the index profile in the core (while retaining a uniform outer cladding) [7,8], or microstructuring [16] can reduce the MFD variation somewhat, but does not eliminate it.

If there was a fiber where the mode field distribution did not change at all with fiber size ($\partial \mathrm{\Psi}/\partial \rho =0$), then according to Eq. (1) the taper gradient $d\rho /dz$ could take any value and the taper could have any length, even zero, and still be adiabatic. We describe such a fiber as being endlessly adiabatic. The only limitation is that the taper should be axi-symmetric, without shear displacements. Loosely speaking, if the fundamental mode remains the same along the taper, the light does not even know it is in a taper rather than a uniform fiber. More precisely, local-mode coupling would be absent in the taper for the same reason it is absent in an ordinary uniform fiber. Even an abrupt taper in the fiber (with a step change in fiber size) would be like a perfectly aligned splice between two fibers with identical modes, with no loss [4].

#### B. Ideal Logarithmic Index Profile

An idealized fiber with a logarithmically graded refractive-index profile $n(r)$ is an example of such an endlessly adiabatic fiber,

We see from Eq. (2) that a change in fiber size $\rho $ along a taper only changes the additive constant ${n}_{0}^{2}+{\mathrm{NA}}^{2}\text{\hspace{0.17em}}\mathrm{ln}(\rho )$. Thus tapering affects the “background” index but has no effect on the variable part of the profile, $-{\mathrm{NA}}^{2}\text{\hspace{0.17em}}\mathrm{ln}(r)$. Since it is the spatial variation of refractive index that determines the mode field distribution, the mode will be independent of the fiber’s scale: $\partial \mathrm{\Psi}/\partial \rho =0$. Therefore, the log-profile fiber is endlessly adiabatic, able to form adiabatic tapers of any length. (The only loss would be a minimal amount of Fresnel reflection due to the changes in background index.) Appendix A investigates this result for adiabaticity criteria other than Eq. (1).

To calculate the fundamental-mode field distribution $\mathrm{\Psi}(R)$ of the log-profile fiber under the weak-guidance approximation, we express the scalar wave equation [17] with the index distribution of Eq. (2) in a normalized form, assuming no azimuthal dependence,

Although there is a quantum-mechanics literature on logarithmic potentials [18–20], we were unable to find exact analytical solutions to Eq. (3). Instead, we solved the equation numerically as described in Appendix B. The resulting field distribution $\mathrm{\Psi}(R)$ is plotted in Fig. 4, together with the field distribution calculated for the step-profile fiber at $\lambda =1550\text{\hspace{0.17em}}\mathrm{nm}$, and scaled so that their MFDs match. The evanescent radius is ${R}_{E}=1.69301$, and the mode field radius (half of the MFD) is ${R}_{\mathrm{II}}=2.00000$. Despite the exotic form of the index distribution, the mode of the log-profile fiber is very similar to that of the step-profile fiber, though a little more peaked in the middle.

Equation (3) contains no parameters, so the field distribution $\mathrm{\Psi}(R)$ expressed using normalized coordinate $R$ is the same for all logarithmic profiles, wavelengths, and taper widths. On an un-normalized $r$ scale, $\mathrm{\Psi}(r)$ depends on $\lambda $ and NA via Eq. (4), but it still does not depend on the taper scale $\rho $. Changing $\rho $ only affects $\beta $, via Eq. (5). For $\lambda =1550\text{\hspace{0.17em}}\mathrm{nm}$, the MFD of the log-profile fiber from Eq. (4),

equals that of the step-profile fiber (10.4 μm) when $\mathrm{NA}=0.095$. In this case, the field distributions would match as in Fig. 4 even on an un-normalized $r$ scale. To quantify how similar the two MFD-matched modes are, we calculated their overlap integral. Expressed in decibels to mimic the loss of an ideal splice between the two fibers [4], the overlap was just 0.015 dB. This is small compared to typical splice losses in practice, showing that the fundamental mode of the log-profile fiber is practically indistinguishable from that of typical telecoms fiber.It is curious that the mode of the log-profile fiber should be so well behaved despite the presence of an unphysical infinity in the index profile at $r=0$. The reason is the mild nature of the infinity of the log function, such that the limit as $r\to 0$ of ${r}^{a}$ ln $r$ is zero for any positive $a$, however small. Thus the profile volume around $r=0$ is negligible and has little effect on the mode.

#### C. Design of a Practical Logarithmic Index Profile

The idealized fiber is unrealistic in two ways. First, the log function is infinite at $r=0$, so the index $n(r)$ must be capped to a finite value near the axis; Fig. 3. This has little effect if the cap region is small enough compared to the mode. For our $\mathrm{NA}=0.095$ example, if the cap diameter is 0.6 μm (less than 6% of the MFD) the effect on the calculated mode is imperceptible on the scale of Fig. 4. The overlap between the calculated modes of the modified and unmodified profiles corresponds to a splice loss of just 0.00004 dB! This is simply a consequence of the previously discussed mildness of the infinity in the log function.

Second, the ideal logarithmic profile fills all space to $r\to \infty $, whereas a real fiber has a finite outer diameter. For mechanical compatibility with the step-profile fiber, we chose the outer diameter to be 125 μm. This has no effect on the mode of the untapered fiber, with its much-smaller MFD of 10.4 μm. However, if the fiber is tapered small enough, its reduced outer diameter starts to influence, then dominate, the guided mode [3,4,12]. The calculated fundamental-mode MFD of a realistic log-profile fiber with the capped index and finite outer diameter discussed above is plotted in Fig. 2, and its field distributions along a taper are given in Fig. 1(b) for comparison with Fig. 1(a). Although constant over most of the range, for diameters smaller than 25 μm the MFD curve starts to follow that of the step-profile fiber, and indeed of a uniform core-free glass rod, as the light becomes guided by the fiber’s outer boundary in all three cases. In this range, we expect the criterion for adiabaticity to limit the taper gradient, since $\partial \mathrm{\Psi}/\partial \rho $ is no longer zero. The practical fiber is therefore not endlessly adiabatic, strictly speaking, when tapered this small. However, this may not be important because tapered fibers are quite resistant to mode-coupling loss once the light fills the cladding anyway [12].

Figure 5 shows our final design of a realistic log-profile fiber designed for splice compatibility (both optical and mechanical) with the step-profile fiber, with $\mathrm{NA}=0.095$ and an outer diameter of 125 μm. Assuming that the fiber would be made from fused silica doped with fluorine (F), the maximum index in the profile is capped to that of undoped silica over a central diameter of 0.6 μm that can barely be seen in the figure. Refractive indices outside the cap region are depressed compared to undoped silica. The total index range across the entire fiber between cap and outer boundary is 0.017, which is well within the capabilities of industrial vapor-phase preform fabrication techniques.

## 3. EXPERIMENT

#### A. Fiber Fabrication and Characterization

Based on the above-mentioned design, we made a suitable F-doped preform using the plasma-activated chemical vapor deposition (PCVD) process [21]. PCVD builds complex preforms from thousands of thin layers, each with its own composition and refractive index. The layers were deposited inside an undoped substrate tube that was mechanically removed afterwards, leaving a preform made entirely from F-doped silica, out to the outer boundary. 1.2 km of fiber with an outer diameter of 125 μm was then drawn from the preform under a draw tension of 40 g weight. An optical micrograph of the fiber’s cross section is shown in Fig. 6(a) alongside a similar image of SMF-28, which is the “step-profile fiber” we used in our experiments (overlooking the fact that its index profile is not actually specified [14] and appears to us to be more complicated).

The experimental index profile in Fig. 5 was measured using the transverse interferometric method (Interfiber Analysis IFA-100), which determines the distributions of both refractive index and stress. The absolute refractive index was referenced to a matching oil rather than fused silica and is unimportant anyway, so in the figure we have shifted the measured profile by $+0.003$ along the vertical axis to coincide with the design profile. The shapes of the design and measured profiles matched closely, and the overlap between the simulated modes of the measured and design profiles at $\lambda =1550\text{\hspace{0.17em}}\mathrm{nm}$ corresponded to a splice loss of $<0.01\text{\hspace{0.17em}}\mathrm{dB}$.

Although we expected the log-profile fiber’s properties to be similar to those of single-mode telecoms fiber, a key difference is that our fiber is not single mode. To observe mode content, we launched light from a 1550 nm diode laser into 2.5 m of the step-profile fiber, the output of which was butt-coupled to 30 m of the log-profile fiber. Near-field light patterns at the output of the log-profile fiber were imaged onto an InGaAs camera using a microscope objective. Examples of such images are shown in Fig. 6(b). When the fibers were aligned, the output was a symmetric pattern like a typical fundamental mode, but when one fiber was offset laterally it was possible to excite other modes. Indeed, it was surprisingly easy to excite patterns that looked like pure higher-order modes.

To reliably characterize fundamental-mode propagation in the fiber and indeed demonstrate that the fiber can replace conventional single-mode fibers in tapered components such as fused couplers, we need to ensure that its higher-order modes can be suppressed. This is one reason why we designed the fiber to be splice-compatible with the step-profile fiber. We spliced 2.5 m of the step-profile fiber to 2 m of the log-profile fiber using a Fujikura 70S fusion splicer. After some practice and a slight reduction in the arc power (F-doped silica has a lower softening temperature than undoped silica [22]), we could repeatedly make splices with an average loss of 0.14 dB. This was measured by launching 1550 nm light into the step-profile fiber and detecting the power emerging from the log-profile fiber.

Although the output of the spliced log-profile fiber looked like a fundamental mode, Fig. 6(b), it is quite possible that some detected light was in higher-order modes. We therefore spliced another 2.5 m length of the step-profile fiber to the output of the log-profile fiber. This does not prevent multimode propagation between the splices, but it does ensure that only a single-mode output is measured. The average loss of the second splice was 0.25 dB. To explore whether higher-order modes were excited in the log-profile fiber, we gently disturbed it by hand in between the splices. If higher modes were present, we would see an output variation due to multi-path interference between the modes [23]. The observed variation of $<\pm 2\%$ indicates that the first splice coupled $<1\%$ of the input light to higher modes that reach the second splice. We can therefore be confident of making substantially single-mode measurements of the fiber. We can also take the 0.2 dB mean of the two splice losses to represent a practical single-mode splice loss between the two fibers.

The attenuation of the fiber was measured using the cutback technique. A piece of the step-profile fiber carrying light from the 1550 nm laser was fusion-spliced (as described above) to the full 1.2 km length of log-profile fiber, and the output power was measured. The log-profile fiber was then cut 30 m beyond the splice, and the new output power was measured. The difference in the outputs corresponded to a fiber attenuation (in the fundamental mode) of $0.60\text{\hspace{0.17em}}\mathrm{dB}/\mathrm{km}$.

We did not study bend loss in detail, but winding 5 turns of log-profile fiber (spliced between two pieces of the step-profile fiber) around a 6.35 mm radius caused less than 0.2 dB of loss at 1550 nm. The loss of the same bend in the step-profile fiber [14] exceeded 30 dB. This is not to say that log profiles are intrinsically superior to step profiles for bend loss—the refractive index contrasts in the two fibers here are different—but it does show that our fiber’s bend loss properties are acceptable as a practical replacement for the step-profile fiber in taper-based components. (The low bend loss of the present fiber contrasts with that of the fiber of [13]. This discrepancy is interesting, and we discuss a possible cause in Appendix C.) With an offset butt-coupled input as shown in Fig. 6(b), one turn of 10 mm radius eliminated all apparent traces of higher modes in the fiber’s output. Bending can therefore be a good way to attain single-mode propagation if a matched-fiber splice is not possible, e.g., at other wavelengths.

#### B. Tapered Fiber

The log-profile fiber was designed to make adiabatic tapers of any length. We tested this by making tapers in a 30 m initial length of the fiber spliced between two pieces of the step-profile fiber. 1550 nm light was launched into one piece of the step-profile fiber, and the output from the other was monitored as tapering proceeded. After each taper was measured it was discarded, and the output splice was re-made, ready for the next taper. We consumed 14 m of the fiber for 20 such measurements.

Our tapering rig was programmed to produce a taper waist of 30 μm diameter and 10 mm length, with an input transition of zero length (as determined by the motion of the motorized stages and assuming a point-like “flame brush” heat source [2]) and an output transition 17 mm long with a linear profile; Fig. 7 inset. Because the heat source (an oxy-butane flame) was not actually point-like, the input taper transition was 1.7–2.0 mm long, reflecting the size of the flame. In other words, this was the shortest taper transition our rig was capable of making with that heat source.

The output transition was much longer because we wanted to measure the loss of just one short transition: the one at the input end of the taper. The presence of two potentially lossy transitions would allow multipath interference between them, complicating the interpretation of our measurements (and indeed disguising the losses if the interference was constructive). Since the loss of a 17 mm long taper in the step-profile fiber was 0.1 dB, the equally long output transition in our tapers should be reliably adiabatic.

The losses of the 20 short tapers made using the log-profile fiber ranged from 0.03 to 0.62 dB, with a median loss of 0.18 dB; Fig. 7. In contrast, the losses of 20 tapers made in exactly the same way using the step-profile fiber ranged from 1.15 to 4.98 dB, with a median loss of 2.29 dB. The scatter in these measurements is likely to be due to imperfections in the tapering process. Inadvertent transverse displacements (e.g., shearing due to slight misalignment of fiber clamps) will have a much greater impact on the losses of very short taper transitions than on the longer ones we usually make on our tapering rig; the resulting kink would correspond to a tighter bend [24]. As discussed in Section 2.A, we only expect the log-profile fiber to be endlessly adiabatic for axi-symmetric tapers. If we assume that such imperfections are randomly present in our samples, then the minimum measured loss of the tapers in each fiber (0.03 dB for the log-profile fiber and 1.15 dB for the step-profile fiber) approaches the loss of an axi-symmetric taper of that profile in that fiber.

Figure 8 shows typical optical micrographs of short tapers made in the fibers. Also shown are plots of the taper shapes extracted from the micrographs by image analysis, assuming that the widest section in each image is 125 μm wide. Although differences in illumination make this procedure unreliable quantitatively, it does confirm that the taper shapes are very similar.

To examine the effect of tapering on the mode size, a taper transition long enough (3.7 cm) to be both adiabatic and repeatedly cleaved was formed in a piece of log-profile fiber spliced to step-profile fiber at its input. The taper was cleaved at various points along the transition, and the near-field pattern emerging from the cleaved end was imaged. This was repeated for a similar taper in step-profile fiber. The images are shown in Fig. 9. The pattern in the log-profile fiber varies very little with fiber diameter, whereas the pattern in the step-profile fiber expands considerably in the narrow end of the taper, as expected from the calculations in Fig. 1.

## 4. DISCUSSION AND CONCLUSIONS

We have shown that a fiber with a mode field distribution that stays the same as the fiber changes size is endlessly adiabatic. This means that axi-symmetric tapers made using the fiber will be adiabatic, with no mode-coupling loss, however short the taper transition may be. Fibers with an idealized logarithmic refractive-index distribution have this property.

Our experimental fiber mimics the idealized fiber while avoiding its unphysical infinities in index and size. Its guidance properties resemble those of standard telecoms fibers, except that it is not single-mode. The fiber is directly splice-compatible with the step-profile fiber SMF-28, it behaves as a single-mode fiber when spliced to the step-profile fiber, it has low bend loss, and it can be tapered shorter than 2 mm with a loss of 0.03 dB (compared to 1.15 dB for a similar taper in the step-profile fiber).

Short lengths of the fiber can therefore be used to make compact low-loss tapered components that are directly compatible with widely used fiber types. We expect it to have applications where it is desirable to make tapered fiber components as short as possible (such as in the manufacture of fused couplers [5]), and where the scalability of tapered components is limited by the need to be adiabatic (such as the number of modes multiplexed in mode-selective photonic lanterns [6–8]).

In the case of photonic lanterns made from bundled fibers, the taper transition has a portion where the light spreads out from the cores to fill the individual fibers. Using log-profile fibers, this portion can have zero length. Then there follows a portion where the light from the individual fibers spreads across the common fused cross section of all the fibers. Adiabaticity requires a finite (non-zero) length for this portion, but that length is shorter using log-profile fibers because the structure is smaller in area at the start [6]; in Fig. 2, the step-profile MFD reaches the outer boundary (the uniform-rod MFD) at a much bigger diameter than the log-profile MFD does.

Other than trivial cases with a uniform outer cladding, this is to our knowledge the first fiber to be made where the entire refractive index distribution (as far as the outer boundary) has been controlled to yield the fiber’s key optical properties. Indeed, grading the profile to the outer boundary will reduce $\partial \mathrm{\Psi}/\partial \rho $ and to some extent help make tapers adiabatic even if the profile is not logarithmic. We suspect this extended grading is also responsible for the low bend loss [25].

The refractive index range available with fluorine-doped silica (corresponding to a numerical aperture of $\sim 0.2$ in a step-profile fiber) was enough to match the mode of SMF-28, with an MFD of 10.4 μm at 1550 nm. To match a step-profile fiber with a smaller MFD but the same 125 μm outer diameter (e.g., for a more strongly confined mode or simply to guide shorter wavelengths), a greater index range is necessary if the central cap region is to stay $\sim 6\%$ of the MFD. Doping the inner regions of the fiber with Ge to raise the index potentially doubles the index range, and the use of other glass systems may allow even greater ranges, though suitable index-grading technologies are most well developed for doped silica. Alternatively, it may be that a cap region greater than 6% of the MFD can be tolerated for a given application. (Our experimental fiber effectively had a bigger cap than the design—see Fig. 5.)

In the appendices we show that the idealized log-profile fiber has an MFD that is proportional to the wavelength, though we have not yet confirmed this experimentally. This means that one fiber can be well-matched to single-mode fibers of given numerical aperture designed for any wavelength. It also means that the far-field diffraction angle is independent of wavelength. Finally, care should be taken not to draw an F-doped log-profile fiber under high draw tension, because the resulting strain-induced index change will be disproportionately concentrated in the middle of the fiber and degrade its properties. This would be less of a problem in a fiber made by doping with germanium, since the least-fluid glass (where stress is concentrated) will lie at the periphery of the fiber.

The data underlying the results in this paper are available at [26].

## APPENDIX A: OTHER ADIABATICITY CRITERIA

The modified “weak power transfer” adiabaticity criterion of Eq. (1) led to our understanding of endlessly adiabatic fibers because it explicitly contains the sensitivity of the mode pattern to the scale of the fiber, $\partial \mathrm{\Psi}/\partial \rho $. However, we can show that the fiber is endlessly adiabatic using the original form of the weak power transfer criterion [6,11] instead,

However, the most well-known adiabaticity criterion is the intuitively derived “length scale” criterion [3,11],

This provides no insight here, unfortunately. It is oversimplified and takes no account of mode field distributions, replacing the informative integral in Eq. (1) with a simple factor of $1/\rho $.## APPENDIX B: CALCULATING THE MODES OF LOG-PROFILE FIBERS

The scalar wave equation of Eq. (3) was solved for eigenvalue ${R}_{E}$ and eigenfunction $\mathrm{\Psi}(R)$ using Wolfram Mathematica’s differential equation solver NDSolve [27] and a shooting method. Wary of the infinite refractive index at $R=0$, we derived an analytical solution valid for small $R$, including $R=0$,

Substitution into Eq. (3) shows this solution to be accurate provided $\mathrm{\Psi}$ stays close to its on-axis value, which we took to be $\mathrm{\Psi}(0)=1$. We used Eq. (B1) to find inner-boundary conditions for $\mathrm{\Psi}$ and $d\mathrm{\Psi}/dR$ at small $R=0.01$, allowing us to start the numerical integration away from $R=0$. (This precaution was probably unnecessary; Eq. (3) is well-behaved in the limit $R\to 0$, where the boundary conditions are simply $\mathrm{\Psi}=1$ and $d\mathrm{\Psi}/dR=0$.) Our outer-boundary condition was $\mathrm{\Psi}=0$ at $R=24.04$, equivalent to the edge of a 125 μm diameter fiber.The normalized Petermann-II mode field radius [15] ${R}_{\mathrm{II}}$ was calculated, as a fixed value of $R$, from the numerical solution for $\mathrm{\Psi}$. The scaling of Eq. (6) shows that the un-normalized MFD is proportional to the wavelength of the light (as required by the fact that the wavelength is the only length scale in the problem) and inversely proportional to NA. It is as if the log-profile fiber behaves at every wavelength like a step-profile fiber designed for that wavelength. Furthermore, because the far-field angle of Fraunhofer diffraction is proportional to wavelength divided by aperture size, the angular size of the far-field pattern from the fiber will be independent of wavelength. It will depend only on the parameter NA—which is why we think of it as a numerical aperture and named it thus.

The fundamental mode for the realistic fiber with a capped refractive index was found similarly, except in this case the field in the cap is the familiar core solution for a step-index fiber, the Bessel function ${J}_{0}$ [17]. We started the numerical integration from the edge of the cap.

The overlap integral between two simulated modes ${\mathrm{\Psi}}_{1}$ and ${\mathrm{\Psi}}_{2}$,

Although the simulation of higher-order modes was not considered in this paper, we give our method of solution here for completeness. For modes with azimuthal order $l=0$ (i.e., ${\mathrm{LP}}_{0,m}$ modes with no dependence on angle $\varphi $), we simply find further solutions of Eq. (3) with greater ${R}_{E}$. For modes with $l\ne 0$ [i.e., ${\mathrm{LP}}_{l,m}$ modes with a $\mathrm{cos}(l\varphi )$ or $\mathrm{sin}(l\varphi )$ angular dependence], the scalar wave equation becomes

It is interesting to note that the logarithmic profile is the $\alpha \to 0$ limit of the infinite (un-clad) power-law profile [17,19],

$B$ and $C$ are independent of $r$, but it is important to allow them to depend on $\alpha $, otherwise the $\alpha \to 0$ limit of Eq. (B5) is trivial. One way to force a non-trivial case is to require the profile to pass through fixed points $n({r}_{1})={n}_{1}$ and $n({r}_{2})={n}_{2}$, with ${r}_{1}<{r}_{2}$ for definiteness and ${n}_{1}>{n}_{2}$ to ensure the profile is a guide rather than an antiguide. This determines $B$ and $C$, and Eq. (B5) becomesEach power-law profile of positive $\alpha $ has an analytical relationship with a partner profile of negative $\alpha \u2019=-2\alpha /(\alpha +2)$ [19,28]; the $\alpha \to 0$ log profile is the median case that partners itself.

## APPENDIX C: EFFECTS OF THE FIBER DRAWING CONDITIONS

The experimental results in this paper were obtained using fiber drawn under a draw tension of 40 g. However, the preliminary results reported in [13] were obtained using an earlier fiber drawn from the same preform under a greater tension of 120 g, because we had been concerned to minimize dopant diffusion. At the time we had not directly measured the fiber’s index profile, just the profile of a separately drawn test fiber. When we did measure it, Fig. 10(a), it was a much worse fit to the design. The index peak was reduced, which broadens the mode field distribution and hence weakens optical confinement. Although the earlier fiber formed low-loss short tapers, its bend loss was poor.

We believe this to be due to residual tensile stresses in the fiber after drawing at high tension. Stress profiles measured by the IFA-100 instrument are plotted in Fig. 10(b), showing a much higher peak stress in the earlier fiber. The stress is concentrated at the center of the fiber, where the least-doped and hence least-fluid glass lies [22]. The stresses have the right sign and order of magnitude to explain the differences between the index profiles. Figure 10(c) shows the index profiles of Fig. 10(a) corrected for the effects of the stress profiles of Fig. 10(b), using the index-stress relation in [29]. Although not identical, the two corrected index profiles (which estimate the index that would result in the absence of stress) are much closer.

Ironically, in due course we realized that a logarithmic dopant distribution is invariant under diffusion anyway; it is the 2D equivalent of a uniform concentration gradient in 1D. We therefore expect moderate diffusion to degrade our fiber’s index distribution only at the center and outer edge. In practice we saw no evidence for significant dopant diffusion in any of our experiments.

## Funding

Science and Technology Facilities Council (STFC) (ST/N000544/1); European Commission (EC) (EU/FP7 312430 “OPTICON”); Engineering and Physical Sciences Research Council (EPSRC) (PhD studentship, KH).

## Acknowledgment

The authors thank J. Harris and W. J. Wadsworth for useful discussions on fiber fabrication.

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