## Abstract

The impact of Higher-Order Mode content on beam quality in large mode area fibers supporting several guided modes is carefully investigated. It is shown that even excellent beam quality (M^{2} < 1.1) in LMA fibers does not guarantee low HOM content, and that the presence of HOMs can lead to significant uncontrollable changes in beam quality, peak intensity, and pointing uncertainty that depend on the uncontrollable relative phase of the modes in the fiber.

©2007 Optical Society of America

## 1. Introduction

Low NA, Large Mode Area (LMA) fiber designs have enabled the recent revolution in high power fiber sources by allowing single-mode-like behavior from fibers with an effective area (A_{eff}) that can be orders of magnitude larger than standard single-mode (SM) fiber designs. Many desirable features of SM behavior can be obtained using LMA fibers supporting several modes by coiling these fibers to exploit the larger bend loss experienced by Higher-Order Modes (HOMs) in comparison to the fundamental mode [1]. This tactic is now routinely used and has led to increasingly impressive peak and average powers from a single fiber while maintaining an output beam quality, as characterized by the M^{2} parameter, that would seem to be suggestive of SM or nearly SM operation [2]. Two obvious limitations exist when scaling this approach to arbitrarily large mode areas. First, the need to introduce large HOM loss while maintaining small fundamental mode loss becomes challenging as fiber dimensions grow. Second, the increasingly small mode spacing for very large fibers allows light to be continually coupled back into HOMs by unavoidable perturbations that are either inherent in fiber fabrication or that occur in any realistic operating environment [3]. There have recently been several very significant steps forward in understanding both the fundamental and the practical constraints that challenge the ability of new fiber designs to offer effectively SM behavior while continuing to increase A_{eff} [4–6], but laboratory demonstrations of LMA fibers now almost exclusively use low M^{2} as a means to imply that SM performance is achieved.

In this paper numerical mode calculations for an exemplary LMA fiber followed by numerical propagation of the free-space output beam are used to explore how HOM content affects beam propagation. It is found that, for the HOMs that are the most difficult to completely strip with bending techniques, a surprisingly large amount of HOM content (e.g. >30%) can be present while still maintaining a beam that is nearly round and has excellent *M ^{2}* (e.g. <1.1). Several undesirable issues arise in such cases in spite of the demonstration of good

*M*because the properties of the light exiting the fiber become sensitive to the relative phases of all the modes present at the fiber output. Since these phases drift with fluctuations in temperature and other environmental factors that are difficult to reliably control, important parameters such as

^{2}*M*, peak intensity, and beam pointing are likely to drift with time in an uncontrolled manner. Propagation in any pure mode, whether the fundamental mode or any single HOM [7], solves these problems, but this work will show that the demonstration of an

^{2}*M*

^{2}value near the ideal

*λ*201C;diffraction limit” of 1.0 for a Gaussian beam is by no means sufficient to verify SM propagation.

_{s,i}## 2. Analysis and results

The exemplary fiber that will be considered here has an ideal step-index profile with a 50 μm core, a 250 μm cladding, and an NA of 0.065. The general conclusions regarding the impact of HOM content on important beam parameters hold for a very wide range of both passive and rare-earth-doped LMA fiber designs, and these particular parameters were chosen only to be representative of an aggressive but commercially-available LMA fiber. A commercial numerical mode solving software package (FiberCad) was used to calculate all 13 guided LP modes for a fiber with the above index profile. In addition to determining the mode profiles, the commercial software was used to estimate the bend loss as a function of bend radius for each, and these results are shown in Fig. 1. Although these bend loss estimates are not expected to be quantitatively accurate, they do reveal properties that also hold qualitatively true for a large design space region around this particular fiber, such as which modes are the most difficult to strip with coiling. Completely extinguishing an HOM using coiling requires significant bend loss for the mode in question, but it also requires that the HOM not be continually repopulated due to coupling between the desired (usually fundamental) mode and the undesired HOM caused by fiber perturbations. Such perturbative coupling out of the fundamental LP_{01} mode and into an HOM is the strongest for the LP_{11} mode [3–7], which in combination with its relatively low bend loss makes this mode the most problematic.

The large deviation of the LP_{11} mode from cylindrical symmetry also leads to the most interesting and the most problematic behavior, and the remaining analysis presented here will therefore consider the set of modal superposition states consisting of different admixtures of the calculated LP_{01} and LP_{11} modes present at the near-field fiber output:

where *A _{LP11}* is the power fraction in the LP

_{11}mode and

*Δϕ*is the relative LP

_{01}-LP

_{11}phase. The propagation properties of the output light are determined by both the fraction of each mode that is present and by the relative phases of these two modes, and the impact of both these parameters was investigated to generate the results that follow. Implicit in writing down this superposition is the assumption that the light propagating in the fiber has sufficiently narrow linewidth that there is a single well-defined phase relationship between the two fiber modes. This issue of coherence will be further discussed at the end of this section.

The various calculated mode profiles at the fiber output were propagated into the far field by using the Fourier Transform relationship between the near and far field profiles of the electric field. Several beam parameters of interest, such as peak intensity and beam centroid, can be calculated directly from the near- and far-field patterns. In addition, if the beam width is defined using the rigorous second moment definition

where the width (radius) is *W*=*2σ*, then *M ^{2}* is defined by the propagation equation

At a location *z* that is in the far field, the near and far field beam widths (radii) are related by

so that the *M*
^{2} parameter can be easily calculated from the near- and far-field beam widths. The minimum possible value for *M*
^{2} of 1.0 is achieved for an ideal Gaussian beam, and *M*
^{2} roughly describes how much faster than the ideal diffraction limit a given beam diverges [8]. Although *M*
^{2} is the metric used most frequently in current LMA fiber literature to characterize beam quality, it is related by a simple λ/π proportionality constant to the Beam Propagation Parameter (radius times half-angle divergence) often used to describe other types of lasers.

For reference, Fig. 2 shows the near and far field profiles for the pure LP_{01} and LP_{11} modes in the fiber under consideration. The indicated spatial positions are in units that are normalized to the near- or far-field beam radius, and the origin of the coordinate system in both the near and the far field is the referenced to the centroid of a pure LP_{01} mode. The *M*
^{2} values for these two pure modes in the x direction are 1.07 and 3.16, respectively, and in the y direction they are 1.07 and 1.05, respectively. Figure 3 shows an exemplary superposition state that consists of 30% LP11 and 70% LP_{01} for the limiting cases of both zero and π/2 relative phase, and it illustrates several important qualitative features of LP_{01}-LP_{11} superpositions. First, the *M*
^{2} value in both cases is never worse than an impressive-looking 1.35, and it can be as good as 1.08 which practically indistinguishable from that of the pure LP_{01} mode even though the LP_{11} content is by no means small. Second, both the near- and far-field profiles are surprisingly round and uniform looking (especially for the in-phase case) in consideration of the well-known asymmetry of the LP_{11} mode and its large fraction. Third, for the in-phase case the near-field centroid is offset by more than half a beam radius although the far-field centroid is not offset at all, and for the π/2-phase case this qualitative situation is reversed. Finally, although the beam quality in all cases would likely be considered quite good despite the large LP_{11} content, it nonetheless changes by a significant fraction with the change in phase.

These qualitative features are further illustrated and quantified in additional figures below that describe beam propagation behavior of the full range of possible LP_{01}-LP_{11} superpositions. Figure 4(a) shows how *M*
^{2} varies with LP_{11} fraction for the limiting cases of 0 and π/2 relative phase, and Fig. 4(b) shows how *M ^{2}* in the x direction varies with relative phase for several examples that span the full range of possible LP

_{11}fractions.

*M*in the y direction is not shown in Fig. 4(b), but it does not appreciably vary with either LP

^{2}_{11}fraction or relative phase. Much beyond the ambiguity about the precise

*M*value sufficient to demonstrate “excellent” beam quality and “nearly” SM performance, these results make clear that a single

^{2}*M*measurement is fundamentally unable to verify that propagation of a single mode in an LMA fiber has been achieved. For LP

^{2}_{11}fractions that approach 30% and zero relative phase, the

*M*value does not vary at all with LP

^{2}_{11}fraction and is indistinguishable from a pure LP

_{01}mode.

*M*can become measurably worse for other phase values (π/2 being the worst), but since the relative modal phase is not under deterministic experimental control and is not measurable or otherwise knowable from only a single

^{2}*M*measurement, one cannot know from a single

^{2}*M*measurement whether the best case or the worst case was observed. Therefore one cannot obtain any significant information about the amount of HOM content from just the measurement of a single good

^{2}*M*value.

^{2}Figure 5 shows how the *x* position of the centroid of the near- and far-field beam varies with phase for different LP_{11} fractions, where the position is displayed in units that are normalized to the near- or far-field beam radius. Because of the choice of x-axis with respect to the LP_{11} symmetry axis, the *y* position of the centroid does not change with LP_{11} fraction or relative phase and is not shown. It can be seen from Fig. 5 that the near- and far-field centroids move by as much as approximately one beam radius with phase changes, and it is particularly noteworthy that beam motions of nearly this size occur for LP_{11} fractions as small as 20%. Since *M ^{2}* can be indistinguishable from the pure LP

_{01}value for LP

_{11}fractions as large as 30%, it is clear that limited measurements showing a good

*M*value do not guarantee the beam pointing stability that would be provided by true single-mode behavior.

^{2}Figure 6 shows how the near- and far-field spatial peak intensity within the beam varies with phase for different LP_{11} fractions, where the peak intensity is normalized to the value for the pure LP_{01} mode in the near or far field. The near- and far-field peak intensities are both seen to vary with phase by as much as a factor of two for some LP_{11} fractions, and it is again particularly noteworthy that peak intensity changes of ∼2x in the near field and ∼1.5x in the far field occur for the ∼30% LP_{11} fraction that an *M ^{2}* measurement could easily confuse with ideal single-mode behavior. It is also interesting to note that for any given LP

_{11}fraction, the far-field peak intensity is maximized for the relative phase value that gives the worst

*M*. This initially counter-intuitive situation is a result of the fact that the near-field profile has a larger size for this phase, so that the far-field spot is smaller even for an

^{2}*M*that is somewhat worse than the “diffraction-limited” value of 1.0.

^{2}Finally, Fig. 7 shows a movie that illustrates how the far-field intensity pattern varies with phase for a fixed LP_{11} fraction of 30%. This view of the changing beam is likely to be familiar to all who have experimental experience with fibers in this class—the process of setting up measurements with such fibers can easily lead to moderate motion of a round and relatively uniform-looking output beam. Since mechanically mounting the end of an LMA fiber in a way that is stable but that does not introduce loss or HOM coupling is in itself a highly non-trivial task, it can be very easy to mistakenly attribute moderate beam motions to mechanical motion of the fiber tip when some part of the measurement apparatus is slightly disturbed. Such beam motion could of course be caused by accidental movement or rotation of the fiber, particularly for angle-cleaved fibers, but it could also be caused by disturbances to the fiber path that alter the relative modal phase. Furthermore, it would be quite possible to attempt an empirical optimization of the fiber mount or packaging by making successive *M ^{2}* measurements, and to thereby obtain a result that has delivered a good

*M*by unintentionally optimizing the relative modal phase rather than by the intended changes that were made to the fiber mount or packaging. These subtleties capture just a few of the issues that challenge the task of making repeatable and accurate LMA fiber beam quality measurements.

^{2}I will conclude this section by returning to the issue of coherence. As stated above, the preceding analysis is based on the implicit coherence assumption that two modes in an LMA fiber interfere at the fiber output with a single well-defined relative phase. This assumption is true only for a sufficiently narrow linewidth source because the relative modal phase at the fiber output depends on wavelength as well as on the effective index difference for the interfering modes and on the fiber length, and the relative modal phase may therefore vary substantially across the spectrum of broad-band optical sources. The linewidth requirement needed for coherent modal interference in the case of a partially coherent source is

where *Δλ* is the source linewidth, *λ* is the center wavelength, *Δn _{eff}* is the effective index difference for the two interfering modes, and

*L*is the fiber length. For the exemplary LMA fiber under consideration in this paper, a typical length for a Yb amplifier fiber operating near 1064 nm is ∼1 m, and

*Δn*between the LP

_{eff}_{01}and LP

_{11}modes is ∼ 10

^{-4}, so the source coherence requirement is that the linewidth be substantially less than 1 nm. Equivalently, for transform-limited pulsed sources, coherent modal interference occurs when the pulse duration is substantially longer than ∼ 1 ps. It is worth noting that many CW and nanosecond sources easily satisfy this coherence condition. Although it will not be discussed in detail here, the behavior of LMA fibers in the incoherent limit is far more intuitive--the centroid of the beam does not move, and the

*M*of the output is a simple weighted average of the

_{2}*M*for each of the populated modes.

^{2}## 3. Implications and conclusion

The very simple, but very important, conclusion of this analysis is that demonstration of excellent beam quality and M_{2} parameter from an LMA fiber source is not sufficient to verify that the source is operating in a single mode. The potential impact of unexpected HOM content is that the relative phase of the propagating modes will in general drift in an uncontrolled way with environmental fluctuations or even with wavelength fluctuations. The fractional coupling into the HOMs of an LMA fiber can be made stable (for example with a well-mounted splice to SM fiber) and is deterministically fixed by overlap of the launch field with the fiber modes, but uncontrollable phase drifts can cause the *M*
_{2} beam quality factor to fluctuate by some significant fraction of its average value, the beam position to shift by a total amount on the order of one beam radius, and the peak intensity to drift by 50% or more. For many applications these issues may be completely insignificant so long as the *M _{2}* is below some threshold value and the system design has margin to assure that a minimum number of photons always reach the target. In addition, applications that require femtosecond or picosecond pulses will not exhibit these coherent interference effects. But LIDAR and other applications that require accurate, spatially-resolved backscatter measurements from nanosecond or longer pulses may be very sensitive to these effects.

A detailed prescription for quantifying the amount of HOM content and for verifying that single-mode behavior has been achieved will be the subject of a future paper, but a good general strategy for obtaining a more complete view of the HOM content in a fiber source involves making a sequence of measurements while varying some parameter that would change the relative phase of the modes in the fiber. For example, the output beam profile could be monitored in real time while the temperature of the fiber is varied so that the associated changes in modal phase will cause the intensity pattern to move or otherwise change in rough proportion to the amount of LP_{11} content. Developing a calibrated relationship between beam deviation and the amount of HOM content may be impossible as a practical matter, but even qualitative information can be a great aid in optimizing fiber performance or in approaching truly SM performance. Scanning the wavelength being transmitted through the LMA fiber will also change the modal phase, and a rough estimate for the exemplary fiber shows that a ∼1 m fiber length would require a wavelength sweep of ∼10 nm to sweep the relative LP_{01}-LP_{11} phase by 2π. If the output of an LMA fiber can be coupled back into a truly single-mode fiber, then propagation of the different modes in the LMA fiber will effectively define the two arms of a Mach-Zender interferometer with the SM fiber as its output. If the throughput is then measured as the wavelength is scanned, or if the transmission of a broad-band source through the LMA fiber is monitored with an optical spectrum analyzer, then the period and modulation depth of the spectral fringe pattern will contain quantitative information about the amplitudes of the modes present in the fiber. This technique is analogous to methods previously used to measure the amount of multi-path interference in HOM dispersion-compensating fibers [9].

The tremendous potential offered by fiber sources for many high power applications is indisputable, but the results presented here suggest some areas that warrant attention when transitioning hero LMA fiber experiments from the lab to the field.

## Acknowledgments

I gratefully acknowledge useful conversations with Doug Holcomb.

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