## Abstract

A new measurement technique, capable of quantifying the number and type of modes propagating in large-mode-area fibers is both proposed and demonstrated. The measurement is based on both spatially and spectrally resolving the image of the output of the fiber under test. The measurement provides high quality images of the modes that can be used to identify the mode order, while at the same time returning the power levels of the higher-order modes relative to the fundamental mode. Alternatively the data can be used to provide statistics on the level of beam pointing instability and mode shape changes due to random uncontrolled fluctuations of the phases between the coherent modes propagating in the fiber. An added advantage of the measurement is that is requires no prior detailed knowledge of the fiber properties in order to identify the modes and quantify their relative power levels. Because of the coherent nature of the measurement, it is far more sensitive to changes in beam properties due to the mode content in the beam than is the more traditional M^{2} measurement for characterizing beam quality. We refer to the measurement as ** S**patially and

**pectrally resolved imaging of mode content in fibers, or more simply as S**

*S*^{2}imaging.

©2008 Optical Society of America

## 1. Introduction

Large-mode-area (LMA) fibers have enabled recent advances in high-power fiber lasers and amplifiers. However, many applications of fiber lasers depend on the quality of the beamprofile. While single-mode fibers (SMF) are known for their excellent beam quality, as the effective area (A_{eff}) is pushed larger to enable high power operation and mitigate nonlinearities, the fiber begins to support increasing numbers of higher-order-modes (HOMs) which can degrade the output beam quality.

A typical measure of the quality of an optical beam is the M^{2} parameter [1]. Frequently a low value of M^{2} is considered to be equivalent to single mode operation with a stable beam. However, even when the amount of power contained in a higher-order-mode becomes very large, it is still possible to achieve a low value of M^{2} [2, 3]. Even worse, changing the relative phase of the modes propagating in the fiber can lead to pointing instabilities in the far field.

In the context of optical communications systems, Multi-Path Interference, or MPI, is a well known impairment caused by the beating of signals and weak, delayed replicas generated, for example, by Rayleigh scattering [4]. In the case of few-mode fibers, coherentMPI results from modes that propagate with different group delays and can lead to signal fading on a slow time scale [5]. This fading is analogous to the beam pointing instability that will occur in an LMA fiber as the phases between modes slowly drift. Furthermore, as the number of scattering sites increases, either due to discrete interfaces such as splices between LMA fibers, or distributed scattering inside the fiber, the impairment due toMPI rapidly increases [6]. Consequently, accurate techniques are required to characterize the MPI of optical beams from LMA fibers capable of supporting multiple HOMs.

The fraction of power of the output of an optical fiber contained in the fundamental mode can be estimated by measuring the launch efficiency of the beam into single mode fiber [7]. Measurements capable of quantifying the partially coherent, transverse mode content from bulk-optic laser resonators only require measuring the transverse beam intensity profiles [8, 9]. The situation where the modes are coherent requires more information, however. Various deconvolution algorithms have been presented for determining the modal power from transverse beam intensity measurements if the modes of the fiber are already known [10–12]. However for accurate analysis the modes must be known precisely. Furthermore without additional constraints, the solution of modal weights can contain ambiguities in the retrieval [12, 13].

In this paper we propose and demonstrate a new measurement technique capable of simultaneously imaging multiple, coherent, higher-order modes propagating in an LMA fiber. Not only can the types of modes and their MPI levels be quantified, but the measured data can be analyzed to provide the level of Poynting vector instability in the beam caused by fluctuating phases between the modes. Because the method is interferometrically based, it is sensitive to even very small fractions of power contained in HOMs. We refer to this technique as ** S**patially and

**pectrally resolved imaging of fiber modal content, or more simply as S**

*S*^{2}imaging.

## 2. Experimental implementation of S^{2} imaging and data analysis

S^{2} imaging is based on the idea that modes propagating in optical fibers can be identified by both the group delay difference which leads to a spectral interference pattern in a broadband source propagating through the fiber, as well as by a distinct spatial interference pattern between the high-order mode and the fundamental mode. Figure 1 illustrates the situation for calculated mode profiles of a 27 *μ*m core diameter fiber with low NA. The intensity profiles for the LP_{01} and LP_{11} are shown in Fig. 1(a) and Fig. 1(b). When the electric fields of these two modes are added together, the resulting spatial interference pattern is shown in Fig. 1(c). For this particular calculation, the LP_{11} was assumed to be 10 times weaker than the LP_{01}. Note that the interference fringe visibility is peaked away from the center of the beam, and the precise shape of the spatial pattern and location of the peak in the interference fringe visibility depends on the MPI of the two modes. It is the spatial dependence of the interference pattern as well as the different group delays for different modes in fibers, which allows for simultaneously imaging multiple modes propagating in the fibers.

The S^{2} imaging setup formeasuring the higher-order-mode content of an LMA fiber is shown in Fig. 2(a). Light from a broadband source is launched into the LMA test fiber. At the exit of the LMA fiber the beam is imaged with magnification onto the cleaved end of a single mode fiber which is coupled into an optical spectrum analyzer (OSA). A polarizer ensures that the polarization states of the modes are aligned on the SMF end-face.

The probe fiber, single-moded at the measurement wavelength, is placed on automated translation stages to move the fiber end in x and y directions perpendicular to the beam propagation direction. The SMF fiber is rastered in x and y, and at each (*x*,*y*) point the optical spectrum is measured. Computer control is used to automate the movement of the single-mode fiber probe and acquisition of the optical spectrum. A typical optical spectrum measured at an arbitrary (*x*,*y*) point is plotted in Fig. 2(b). If two different modes overlap spatially at that (x,y) point, they will have a spectral interference pattern due to group delay differences between the modes in the fiber under test. In the plots of the Fourier transform in this work, the x axis of the Fourier transformis scaled by the fiber length to obtain the group delay difference between themodes in units of ps/m. The Fourier transform of the optical spectrum, plotted in Fig. 2(c) shows several different mode beats at different group delay differences. It is this spatially and modally dependent spectral interference pattern that is used to simultaneously image quantify the relative power levels multiple HOMs propagating in the LMA fiber.

The multi-path-interference(MPI), is defined as the ratio of powers P_{1} and P_{2} of the modes, *MPI*=10log[*P*
_{2}/*P*
_{1}]. In order to calculate the relative power levels of the modes using the Fourier transform of the measured optical spectra, we assume two modes with spatially and frequency dependent amplitudes, *A*
_{1}(*x*,*y*,*ω*) and *A*
_{2}(*x*,*y*,*ω*), related by a constant *α*(*x*,*y*), such that

where *α*(*x*,*y*) is assumed to be independent of wavelength. If the group delay difference between the modes is assumed to be independent of frequency, then the spectral intensity caused by interference between the two modes can be written as

where *τ _{b}* is the period of the beat frequency between the two modes caused by their relative group delay difference. The Fourier transform of the spectral intensity is then

where *B*
_{1}(*x*,*y*,*τ*)=𝓕{*I*
_{1}(*x*,*y*,*ω*)} is the Fourier transform of the optical spectrum of a single mode. Although the optical spectra in this work are plotted with respect to wavelength, the Fourier transforms are taken with respect to frequency.

The spatially dependent Fourier transform of the optical spectrum is then used to calculate *α*(*x*,*y*). At a given (*x*,*y*) point, the ratio *f* (*x*,*y*) is defined as the amplitude of the Fourier transform of the spectral intensity at the group delay difference of interest divided by the amplitude at group delay zero. Assuming that the width of *B*
_{1}(*x*,*y*,*τ*) is small compared to *τ _{b}* (see for example Fig. 2(c)),

*f*(

*x*,

*y*) can be written as

Thus, *α*(*x*,*y*) is simply related to *f*(*x*,*y*) by

Because the wavelength range over which the measurement is made covers many beat periods, the total intensity of the two modes measured in the OSA at a given (*x*,*y*) point integrated over the measurement bandwidth is just the incoherent sum of the individual mode intensities. The intensities of modes then are given by

where *I _{T}*(

*x*,

*y*)=

*I*

_{1}

*x*,

*y*)+

*I*

_{2}(

*x*,

*y*) is the integrated optical spectrum at a given (

*x*,

*y*) point. Therefore, at each (

*x*,

*y*) point,

*I*

_{1}(

*x*,

*y*) and

*I*

_{2}(

*x*,

*y*) can be calculated via Eq. 5 and Eq. 6, and then the total MPI calculated from

## 3. Experimental results of S^{2} measurements

For these experiments a 20 m length of LMA fiber with 0.065 NA and 27 *µ*m core diameter was characterized. Taking into account the 13 cm bend radius that was used when coiling the fiber, the A_{eff} was approximately 370 *µ*m^{2} [14,15]. The broadband optical source used in these experiments was an Yb ASE source that was first polarized and then amplified. The single mode output fiber of the Yb ASE source was spliced with a mode-matching splice to the LMA fiber. Figure. 3(a) shows the beam profile obtained by integrating the measured spectrum at each (x,y) point. A sum of the Fourier transforms of all the measured optical spectra from each spatial point is shown in Fig. 3(b). There are clearly several different beat frequencies visible, corresponding to interference between the primary LP_{01} mode and different higher order modes. Because the HOMs are weak compared to the LP_{01} mode, interference between two different HOMs is considered negligible.

Figures 3(c) through (f) show the result of the data analysis described in Section 2 applied to the various peaks observed in the Fourier transform of the optical spectra in Fig. 3(b). A variety of higher order modes were observed, clearly identifiable by their spatial patterns as LP_{11}, LP_{12}, LP_{21}, and LP_{02}. The strongest higher-order-mode, the LP_{11}, was 21.7 dB weaker than the LP_{01}.

The peaks of the Fourier transform were identified by the mode image obtained from the data analysis. In order to confirm the mode identification, the measured index profile of the LMA fiber was used to calculate the expected group delays of the modes, and the group delay difference at which the beat frequencies are expected. The expected beat frequencies obtained from the calculation between the LP_{01} and the various higher order modes are indicated in Fig 3(b) by dashed lines. The beat frequencies obtained from the calculation based on the measured index profile agree well with the identification of the beat frequencies based on the obtained mode images.

In order to ascertain the accuracy of theMPI levels obtained from the calculation of the mode images, a higher-order mode fiber with a long-period grating (LPG) designed to couple from the LP_{01} mode to the LP_{02} mode was characterized. The transmission loss of the LP01 through the LPG is approximately equal to the LP_{02}/LP_{01} MPI. The measured transmission loss of the LP_{01} through the LPG is shown in Fig. 4(a).

After the LPG was characterized with the loss measurement it was then measured using the S^{2} imaging setup. Becuase of the frequency dependance of the LPG, the mode content was characterized at 1055 nm with a 10 nm bandwidth in the OSA, where the LPG spectrum was relatively flat. The beam profilemeasured after the LPG was an LP_{02}, as shown in Fig. 4(b). The Fourier transform of the optical spectra and the modes obtained that correspond to the peaks observed in the Fourier transformare shown in Fig. 4(c). The traditional loss measurement gave an MPI value for the LP_{01} mode of -17.5 dB at 1055 nm. This number compared very well with the value of -18.3 dB obtained from the S^{2} measurement. In addition to the LP_{01} MPI the S^{2} image also showed -31.4 dB of the LP_{11} was excited by the LPG, which was not observed by the simple loss measurement of the LP_{01}. Consequently the S^{2} imaging not only compared very well with the loss based measurement of MPI, but also gave additional information unavailable from the more traditional technique.

## 4. Beam pointing instabilities due to higher-order modes

The S^{2} imaging data is capable of measuring the number, type, and MPI of the higher order mode propagating in a large-mode area fiber. Because these modes are coherent, they beat together and cause changes in beam shape and lead to beam pointing instabilities as the phases between the various modes drift. The implications of higher order mode content on beam pointing instabilities were calculated theoretically in Ref. [3]. In addition, it was pointed out in [3] that the traditional measure of beam quality, M^{2}, is relatively insensitive to such impairments, and low values of M^{2} can be obtained even for a large fraction of power in HOMs.

Because the S^{2} imaging data contains information about all of the higher-mode content propagating in the fiber, the effect of phase changes between the modes can be extracted from the S^{2} data set be analyzing the data in an alternate way. Rather than analyzing the data in the Fourier domain, it can be appreciated that a unique beam profile is being measured at each wavelength, and because the different modes in the fiber have different group delays, varying the measurement wavelength effectively samples possible relative phases between the modes. Analyzing the data in this manner allows the MPI levels of the higher order modes calculated in the spectrum’s Fourier domain to be correlated with the level of Poynting vector instability in the spatial domain. The level of MPI in few mode fibers has been previously measured by measuring the change in interference versus wavelength [5]. However, such an approach underestimated the MPI. Consequently the measurement of beam pointing instabilities obtained in this manner could also underestimate the impact of drift in mode phases and only provides a lower bound on movement of the beam center of mass.

Figure 5(a) shows a movie of the measured beam profile versus wavelength for the 27 *µ*m core diameter fiber. This movie was generated from the same data used to calculate the mode images shown in Fig. 3. Even though the MPI of the individual higher-order modes was measured to be less than -20 dB, significant changes in the beam center of mass and width are observed. Using a narrowband (<1nm FWHM) signal at 1055 nm, the measured M^{2} was 1.06.

Using the data from Fig. 5(a), the change vs. wavelength of the beam’s center-of-mass (COM) and beam diameter (twice the second moment of the intensity profile) can be obtained and are plotted as red curves in Fig. 5(c) and Fig. 5(d) respectively. The COM and beam diameter were both normalized to the beam diameter obtained using the full integrated optical spectrum. Even with an M^{2} value which is effectively diffraction limited, the beam COM was observed to move by as much as 15% of the beam diameter.

A second fiber with similar core diameter was also measured with the S^{2} imaging setup, and was found to have -14.8 dB MPI in the LP_{11} mode, with lower levels of MPI for modes such as the LP_{02} and LP_{21} that were similar to the first fiber. The movie of the beam profile vs. wavelength is shown in Fig. 5(b), and the COM movement and beam diameter vs. wavelength are also plotted as blue curves in Fig. 5(c) and Fig. 5(d). The data in Fig. 5(c) and 5(d) have been offset horizontally for readability. In this case the movement of the beam was dominated by side-to side motion due to the strong LP_{11}, with the COM moving by as much as 40% of the beam diameter. However, the measured M^{2} of this fiber using a narrow bandwidth signal at 1060 nm still had a value of 1.2. Although the distortions to the beam shown in Fig. 5 are observed versus wavelength, it should be emphasized that any perturbation to the fiber that causes a change in the relative phases of the modes propagating in the fiber will cause a similar movement in the beam. Such large movements of the COM, for beams with what would traditionally be considered very good M^{2} values, could have significant impact on experiments that depend on beam pointing stability.

In order to ascertain the level of stability of the S^{2} imaging setup, a single-mode fiber was imaged. Ideally a single mode fiber should show no changes in COM or beam diameter vs. wavelength. The movement of the beam COM and change in beam diameter vs. wavelength are also plotted as black curves in Fig. 5(c) and Fig. 5(d) for comparison purposes. There was some small amount movement of the COM with wavelength, which is likely due to chromatic aberrations or lens mis-alignments in the imaging setup, however the size of this movement is very small compared to that observed due to the presence of HOMs in the LMA fibers. In addition changes in the beam diameter were also very small. These measurements confirmed the stability of the setup and that the changes in beam profile for the LMA fibers were due to the presence of higher-order modes.

## 5. S^{2} imaging of a 44 *μ*m core-diameter fiber

S^{2} imaging was also performed on a 15 m length of a fiber with a 44 *µ*m core diameter and NA of 0.07. The fiber was again measured at 1050 nm using the Yb ASE source. Given the 10 cm bend radius of the fiber coil, the A_{eff} was approximately 700 *µ*m^{2}. The beam profile obtained from the fiber is shown in Fig. 6(a), and the Fourier transform of the optical spectrum is shown in Fig. 6(b). The number of peaks in the Fourier transform is substantially more than that observed with the 27 *μ*m core diameter fiber. The modes corresponding to some of the observed peaks and their MPI levels are shown in Fig. 6(i) through 6(j). Given the complicated structure of the Fourier transform for this fiber, it would likely be very difficult to identify modes based only on a measurement of the optical beat frequencies, and calculations of the group delays of the various modes. However the S^{2} imaging returns clear mode images, even with many different modes propagating in the fiber, because of the discrimination available in both spatial and frequency domains.

A movie of the beam profile variation versus wavelength is shown in Fig. 7(a). The corresponding changes in COM and beam diameter vs. wavelength are plotted in Fig. 7(b) and 7(c). With a narrowband signal, this fiber had a measured M^{2} value of 1.2. For comparison purposes the measurements of the 27 *µ*m core diameter fiber with an M^{2} of 1.2 are also shown in Fig. 7.

The 44 *µ*m core diameter fiber had a low M^{2} and the MPI of all the individual modes was less than -20 dB. By traditional measures this fiber could be considered to have a single-moded output beam. However, changing the relative phases of these weak HOMs was still sufficient to move the COM of the beam by 60% of the beam diameter and cause the beam diameter to change by 35%. In addition, it is interesting to compare the nature of the motion of the of the 27 *µ*m core diameter fiber, which was dominated by a strong LP_{11} mode, and consequently led to mostly side to side pointing instabilities, to the 44 *µ*m core diameter fiber where no single mode was strong.

## 6. Discussion and conclusions

Because *S*
^{2} imaging depends on developing a relative group delay between different modes propagating in the fiber, the requirements on the length of fiber that can be characterized and the bandwidth of the optical source depend on the specifics of the fiber. For example, since the group delay differences between modes in conventional LMA fibers are on the order of 1 ps/m, 5 to 20 m of fiber length is sufficient, and an optical bandwidth of 10 to 50 nm can capturemany beat periods between the modes. The group delay differences for the HOM fiber in Fig. 4(c) however were much larger, and half a meter of fiber was sufficient to build up significant group delays. Furthermore, themeasurement is not overly sensitive to the shape of the power spectrum of the optical source, as long as the assumption of a slowly varying spectrum in frequency is valid, allowing for the derivation of Eq. 4.

As is obvious from Fig. 6, a large number of modes can be characterized simultaneously. The maximum number of modes that can be characterized at once will depend on the resolution of the OSA being used, as well as the extent of the group delay differences between the modes. Also important is the assumption that only one mode was strong, with the other modes being weak in comparison. If two modes are strong, those two strong modes could be characterized using the procedure outlined in Section 2, however the procedure would break down when characterizing weaker modes in the same fiber. Under such conditions, a more sophisticated analysis of the Fourier transform of the optical spectrum would be required.

In addition to introducing a new measurement technique for characterizing propagating modes in LMA fibers, these experiments highlight the importance of measuring the modal content in order to determine the quality of beams from large-mode area fiber lasers and amplifiers. Because of the coherent nature of the modes produced at discrete interfaces such as fiber splices, even low amounts of MPI are sufficient to cause changes in beam pointing and diameter due to fluctuations in the phases between the modes.

It must be emphasized that the beam profiles observed in these experiments were very stable, both for broad bandwidth operation, where the broad bandwidth input effectively averaged over many interference periods between the modes, as well as for the narrow bandwidth signals used during the M^{2} measurements. The stability can be attributed to both the all-fiber nature of the setup, ensuring a stable launch into the LMA fiber, as well as the fact that all the various modes travel through a common path interferometer. Consequently, in a lab environment even slow measurements such as M^{2} can be performed repeatedly and return the same value as perturbations to the fiber are minimized. Only when the fiber is mechanically disturbed in a small way, or subject to temperature variations, does the nature of the changing phases between the HOMs become apparent.

Furthermore, M^{2} is insufficiently sensitive to impairments caused by HOMs, and a low value of M^{2} does not guarantee single-moded operation. Even for a diffraction limited M^{2} value of 1.06, the center of mass of the beam was observed to move by as much as 15% of the beam diameter, and for a still very respectable value of 1.2, the beam moved by as much as 60%of the beam diameter, which could be critical depending on the applications intended for a high power laser based on such a fiber. If a low value of M^{2} does not guarantee single-moded operation, then an interesting corollary is that a high value of M^{2} does not guarantee multi-moded operation. For example, a pure LP_{02} mode such as that shown in Fig. 4(b) has an M^{2} of 3, but in the absence of other higher order modes, the beam would be free of pointing instabilities. A low M^{2} is neither a necessary nor sufficient condition for demonstrating single-mode operation of a fiber laser or amplifier.

In summary, S^{2} imaging is a new technique which provides a wealth of detailed information on modal content unavailable from a simple M^{2} measurement. By acquiring both spectrally and spatially resolved data, modal analysis can be performed without requiring prior knowledge of the fiber properties. The sensitive measurement is capable of quantifying the number, type, and relative power levels of higher-order-modes simultaneously propagating in a large-mode-area fiber. By analyzing the data in different domains, one can correlate theMPI levels of the HOMs with their impact on beam pointing instabilities and changes in beam shape due to fluctuations in phases between the modes. This technique can be usefully applied to wide variety measurements such as LMA component characterization, bend loss measurement of high-order-modes, differential gain measurements of modes in large-mode-amplifiers, and LMA fiber splice optimization.

## Acknowledgments

The authors thank J. Jasapara, and C. Headley for helpful discussions.

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