# Cross-correlated (C2) imaging of fiber and waveguide modes.

**ABSTRACT** We demonstrate a method that enables reconstruction of waveguide or fiber modes without assuming any optical properties of the test waveguide. The optical low-coherence interferometric technique accounts for the impact of dispersion on the cross-correlation signal. This approach reveals modal content even at small intermodal delays, thus providing a universally applicable method for determining the modal weights, profiles, relative group-delays and dispersion of all guided or quasi-guided (leaky) modes. Our current implementation allows us to measure delays on a femtosecond time-scale, mode discrimination down to about - 30 dB, and dispersion values as high as 500 ps/nm/km. We expect this technique to be especially useful in testing fundamental mode operation of multi-mode structures, prevalent in high-power fiber lasers.

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**ABSTRACT:**We implement cross-correlated imaging in the frequency domain (fC2) in order to reconstruct different modes propagating in a multi-mode optical fiber, and measure their relative powers. Our system can complete measurements in under a second (950 ms), with a maximum signal to noise ratio of 25 dB. The system is capable of group-delay temporal resolution as high as 720 fs, and this number can be tailored for a variety of modal discrimination levels by choice of apodization functions and effective bandwidths of the tunable source we use. Measurements are made on a double-clad test fiber to demonstrate simultaneous reconstruction of six guided modes. Finally, the system is used to optimize alignment into the fiber under test and achieve mono-mode purity > 95%, underscoring the utility of fC2 imaging for near-real-time modal content analysis.Optics Express 09/2014; 22(19). · 3.53 Impact Factor - Fanting Kong, Guancheng Gu, Thomas W Hawkins, Joshua Parsons, Maxwell Jones, Christopher Dunn, Monica T Kalichevsky-Dong, Stephen P Palese, Eric Cheung, Liang Dong[Show abstract] [Hide abstract]

**ABSTRACT:**Quantitative mode characterization of fibers with cores much beyond 50µm is difficult with existing techniques due to the combined effects of smaller intermodal group delays and dispersions. We demonstrate, for the first time, a new method using a matched white-light interferometry (MWI) to cancel fiber dispersion and achieve finer temporal resolution, demonstrating ~20fs temporal resolution in intermodal delays, i.e. 6µm path-length resolution. A 1m-long straight resonantly-enhanced leakage-channel fiber with 100µm core was characterized, showing ~55fs/m relative group delay and a ~29dB mode discrimination between the fundamental and second-order modes.Optics Express 06/2014; 22(12):14657-14665. · 3.53 Impact Factor - Thomas T. Alkeskjold, Marko Laurila, Johannes Weirich, Mette M. Johansen, Christina B. Olausson, Ole Lumholt, Danny Noordegraaf, Martin D. Maack, Christian Jakobsen[Show abstract] [Hide abstract]

**ABSTRACT:**In recent years, ultrafast laser systems using large-mode-area fiber amplifiers delivering several hundreds of watts of average power has attracted significant academic and industrial interest. These amplifiers can generate hundreds of kilowatts to megawatts of peak power using direct amplification and multi-gigawatts of peak power using pulse stretching techniques. These amplifiers are enabled by advancements in Photonic Crystal Fiber (PCF) design and manufacturing technology. In this paper, we will give a short overview of state-of-the-art PCF amplifiers and describe the performance in ultrafast ps laser systems.Nanophotonics. 11/2013; 2(5-6).

Page 1

Cross-correlated (C2) imaging

of fiber and waveguide modes

D. N. Schimpf, R. A. Barankov, and S. Ramachandran∗

Photonics Center, Department of Electrical and Computer Engineering,

Boston University, 8 Saint Mary’s Street, Boston, MA 02215, USA

∗sidr@bu.edu

Abstract: We demonstrate a method that enables reconstruction of wave-

guide or fiber modes without assuming any optical properties of the test

waveguide. The optical low-coherence interferometric technique accounts

for the impact of dispersion on the cross-correlation signal. This approach

reveals modal content even at small intermodal delays, thus providing a

universally applicable method for determining the modal weights, profiles,

relative group-delays and dispersion of all guided or quasi-guided (leaky)

modes. Our current implementation allows us to measure delays on a

femtosecond time-scale, mode discrimination down to about - 30 dB, and

dispersion values as high as 500 ps/nm/km. We expect this technique to

be especially useful in testing fundamental mode operation of multi-mode

structures, prevalent in high-power fiber lasers.

© 2011 Optical Society of America

OCIS codes: (060.2310) Fiber optics; (060.2270) Fiber characterization; (060.2320) Fiber op-

tics amplifiers and oscillators.

References and links

1. D.J.

status

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2. D.N. Schimpf,J. Limpert,

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3. L. Dong, H. A. Mckay, A. Marcinkevicius, L. Fu, J. Li, B. K. Thomas, and M. E. Fermann, “Extend-

ing effective area of fundamental mode in optical fibers,” J. Lightwave Technol. 27, 1565–1570 (2009).

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cello, ”Light propagation with ultra-large modal areas in optical fibers,” Opt. Lett. 27, 1797–1799 (2006).

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5. A. Galvanauskas, M. C. Swan, and C.-H. Liu, “Effectively single-mode large core passive and active fibers with

chirally coupled-core structures,” paper CMB1, CLEO/QELS, San Jose (2008).

6. F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. T¨ unnermann, “High aver-

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4 July 2011 / Vol. 19, No. 14 / OPTICS EXPRESS 13008

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11. P. Nandi, Z. Chen, A. Witkowska, W. J. Wadsworth, T. A. Birks, and J. C. Knight, “Characterization of a

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1.Introduction

High-performance, high-power fiber-lasers [1,2] demand excellent output beam-qualities. The

design of novel fiber platforms to achieve this requires the propagation of only one mode in

fibers that are not strictly single-moded [3–7], putting a high premium on modal discrimination.

Therefore, it is essential to develop experimental methods that reveal the modal content and

modal weights. Specifically, techniques that make no assumptions about the fiber under test are

of interest. These techniques rely on the fact that fiber modes travel at different group velocities.

In the case of large mode area (LMA) fibers the relative intermodal group-delays are typically

on the order or less than a picosecond per meter.

Apromisingtechniquecalledspatiallyandspectrallyresolved(S2)imaginghasbeenrecently

demonstrated [8,9], and relies on spectral interference between the different modes of the fiber.

Alternatively, optical low-coherence interferometry may be used, which relies on interference

between an external reference beam and the fiber modes of interest [10,11]. This technique is

potentially more flexible because it does not demand specific group-delay or dispersive charac-

teristics from the primary mode of interest. Previous experiments analyzed the high-frequency

oscillation present in the cross-correlation signal, which requires delaying the reference beam

by fractions of a wavelength. This implementation makes the method not only time consuming

for long intermodal group-delays but also susceptible to fluctuations of the path-length during

data acquisition.

In this contribution, we demonstrate modal reconstruction at small intermodal delays using

the technique of optical low-coherence interferometry. Since it is based on cross-correlations

of the fiber output with a reference beam we will refer to this beam-analysis method as cross-

correlation imaging, or more simply as C2imaging. In contrast to previous data analysis of

optical low-coherence interferometry, we analyze the slowly varying envelope of the signal and

not the (unstable) high frequency oscillation. This slowly varying envelope contains much more

information especially related to chromatic dispersion [12]. C2imaging determines the correct

modal weights, profiles, relative group-delays, and dispersion of all the modes without making

any assumptions about the optical properties of the fiber under test. Particularly attractive is

the fact that a measurement of modal group-delays in LMA fibers as low as femtosecond time-

scales is possible by designing a dispersion balanced interferometer.

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2.Experimental implementation

Figure 1 shows a schematic of the Mach-Zehnder interferometer that is used for the optical low-

coherence interferometry. The fiber under test is placed in the probe arm of the interferometer.

In the reference arm, a computer-controlled translation stage scans across the temporal delay

of each individual mode in the probe arm. At each position of the delay stage, an image of the

interference between the near-field of the fiber output and the collimated expanded beam of

the reference arm is taken with a camera. In this way, at every pixel, the cross-correlation trace

between the reference field and the different modes can be detected. In the reference arm both

the fiber input and the coupling lens are situated on a motorized translation stage moving along

the direction of the collimated input beam. In this way, the beam profile at the output remains

stable, i.e. beam-walking, which would be present for free-space delay stages, is avoided.

beam-

spli?er

delay

stage

fiber

under test

reference arm

probe arm

beam-

combiner

camera

SLD

band-

pass

dispersion

compensa?ng

SM fiber

images

Delay [a.u.]

Cross-correla?on

signal [a.u.]

Delay

mode 1 mode 2

Fig.1.Schematic oftheexperimental setup(SLD:superluminescent diode),andillustration

of the cross-correlation trace expected at one pixel of the stack of images.

Depending on the magnitude of the chromatic dispersion of the mode we are interested in, we

employ a reference arm that is (nearly) dispersion-less (free-space) or that contains a fiber with

a known amount of dispersion. We verify the versatility of this technique by deploying these

modificationstoawidevarietyoffibersrangingfromLMAfibers,withsmallintermodalgroup-

delays, to higher-order mode (HOM)-fibers, with large intermodal dispersion values [13].

3.Data analysis

To reconstruct the modes from the stack of images that are acquired with the camera, we de-

velop a model for optical low-coherence interferometry. The model describes the influence of

dispersion on the cross-correlation trace. This enables us to design experimental configurations

that exhibit better temporal resolution. We can also obtain the modal content after correcting

for the impact of dispersion - a feature that is a distinct advantage over self-interferometric

techniques (e.g. S2imaging).

At a given delay-stage position, the following (temporally averaged) intensity image is cap-

tured with the camera

I(x,y) =

+ΔT/2

?

−ΔT/2

dt|E(x,y,t)|2=

+∞

?

−∞

dω

2π|E(x,y,ω)|2,

(1)

where (x,y) defines the position of the pixel at the two-dimensional camera, and ΔT stands for

the exposure time of the camera. The starting point for the following analysis is the representa-

tion in frequency domain.

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The electric field in frequency domain, E(ω) =?dteiωtE(t), is given by a superposition of

E(x,y,ω) =∑

m

The sum describes the superposition of the fiber-modes. The real modal weights are denoted by

αm. The near-field image of the transverse modes in the detector plane are given by em(x,y,ω).

Their mode-propagation constants in the fiber are βm(ω). Am(ω) stands for the spectral shape

of the amplitude of the m-th mode. For simplicity, we will assume that all modes exhibit the

same spectral excitation (this may not be the case if one particular mode undergoes spectral

filtering, e.g. due to a long period grating), and that the spectrum is the same in the reference

arm (i.e. the fiber does not cause spectral filtering for the overall spectrum), so that A(ω) =

Am(ω) = Ar(ω). In this case, the spectrum is given by S = |A(ω)|2. After propagation of the

modes in the test fiber of length L, their acquired phases are given by

the field coming out of the fiber and the reference field:

αmem(x,y,ω)Am(ω)eiφm+er(x,y,ω)Ar(ω)eiφr.

(2)

φm= βm(ω)·L.

(3)

The last term in Eq. (2) describes the light propagation in the section of the reference arm

which is needed to balance the optical path lengths of the different modes of the probe arm.

The (relevant) phase of the reference field is given by

φr= βr(ω)·Lr+ω

It consists of two contributions: the propagation in free-space path of length d and propagation

in a dispersion compensating single mode fiber of length Lrexhibiting a propagation constant

βr(ω). Particularly, the free-space path d is varied by the delay stage in order to scan across

the different delays of the modes of the test fiber. The transverse electric field of the collimated

reference beam at the detector is er(x,y,ω). The transverse phase-profile at the detector is

assumed to be flat - a condition easily verified by ensuring a sharp-contrast, near-field image of

the output facet of the fiber under test.

By substituting the electric field of Eq. (2) into Eq. (1), we arrive at the expression for the

intensity that contains a background term I0and the term Iint, which is due to interference

between the reference field and the individual modes:

cd,

(4)

I(x,y) = I0(x,y)+Iint(x,y),

(5)

with the two terms written explicitly as

?

I0=

+∞

?

−∞

dω

2π

|erAr(ω)|2+∑

m

|αmemAm(ω)|2+∑

(m?=m?)

2Re

?

αm?e∗

m?A∗

m?(ω)αmemAm(ω)ei(φm−φm?)??

(6)

Iint=∑

m

+∞

?

−∞

dω

2π2Re

?

e∗

r(x,y,ω)A∗

r(ω)αmem(x,y,ω)Am(ω)ei(φm−φr)?

.

(7)

The term I0(x,y) is independent of the delay stage position d. To analyze the data, the

term Iint(x,y) of Eq. (7) is of particular importance. This term can be written in a more con-

venient form by making a few assumptions: Since optical low-coherence measurements are

typically performed with spectra of widths of a few nanometers, the transverse electric fields

are assumed to be independent of frequency, e(x,y,ω) ≈ e(x,y,ω0). Furthermore, the phase-

difference Δφmr= (φm−φr) in Eq. (7) is Taylor-expanded around the center angular frequency

ω0in terms of the angular frequency difference Ω = ω −ω0as follows

Δφmr= Θmr−(τ −τmr)·Ω+Δϕmr(Ω),

(8)

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where the phase-mismatch between the two arms is Θmr= (β(0)

variable τ is defined as d/c, τmrstands for a group-delay difference τmr= (L/vgr,m−Lr/vgr,r),

and the dispersion mismatch between the two arms of the interferometer is described by Δϕmr=

∑k≥2(β(k)

propagation constant β.

With these approximations the term Iint(x,y) of Eq. (7) can be written as follows

m ·L−β(0)

r

·Lr−τ ·ω0), a delay

m ·L−β(k)

r

·Lr)Ωk/k!, where the β(k)stand for the Taylor-coefficients of the mode-

Iint(x,y) =∑

m

Im(x,y,τ) =∑

m

2αmRe[e∗

r(x,y,ω0)·em(x,y,ω0) cmr(τ −τmr) exp(iΘmr)],

(9)

where the cross-correlation function cmris given by

cmr(t) =

1

2π

?

dΩ S(Ω) exp(iΔϕmr(Ω)) exp(−iΩ·t).

(10)

It describes the impact of the frequency-dependent mode-propagation constant on the signal.

Particularly, the group-delay τmrdetermines the position of m-th mode in the signal (as illus-

trated in Fig. 1), and its shape is influenced by group-delay dispersion, Δϕmr(Ω), as well as

the shape of the input spectrum S(Ω). It is worth noting that for a Gaussian spectrum and

a parabolic approximation of the phase Δϕmr(Ω) the integral in Eq. (10) can be analytically

calculated (see Appendix).

For a smooth spectrum, the phase of the integral in Eq. (10) is slowly varying as a function

of delay τ compared to the phase Θmr, so that a decomposition in envelope and fast oscillat-

ing function is possible. In general, the phase of integral Eq. (10) can not be obtained in an

analytical form (for the special case of a Gaussian spectrum and a parabolic phase we provide

an analytical expression in the appendix). We introduce a phase-term ψ that describes the fast

oscillating behavior. This results in a simplified expression for the Im(x,y,τ) in Eq. (9), and

thus, Eq. (5) can be written:

?

The normalized intensity distribution for the reference and m-th mode are given by ir(x,y)

and im(x,y), respectively. The envelope of the cross-correlation traces is described by

|cmr(τ −τmr)|. Specifically, it contains information about the dispersive properties of the

modes. For fiber modes with distinct dispersive behavior, the peaks present in the envelope

will be different for each mode. This reveals the strategy to account for the impact of dispersion

on the modal content.

Equations (11) and (10) form the basis of our data analysis. At first, from the stack of images

we pick data corresponding to the offset I0(x,y) only. The resulting term |I(x,y)−I0(x,y)| is

integratedoverall(x,y)pixels.Thisallowsustoobtainaone-dimensionalsignalasafunctionof

the translationstage position fromwhich wecan determine the envelope ofthe cross-correlation

trace. The locations of the peaks in this signal correspond to different delays (i.e. τ = τmr) that

the modes have experienced as a result of the propagation in the test fiber. From these values

the relative group-delays of the modes can be obtained. The shape of the peaks also provides

informationaboutthedispersionofthemodes:Thefittingof|cmr(τ −τmr)|,usingthemeasured

spectrum S(Ω), around each peak of the experimental envelope data gives the group-velocity

dispersion of each mode. In this procedure, we typically assume that there is an negligible

effect or cancellation of dispersion slope over the source bandwidth, and therefore, consider

only a parabolic phase Δϕmr(Ω). Given these global parameters, the mode profiles are finally

determined by a least-squares fit of the analytical model to the envelope data at every (x,y)

coordinate. As a result, a map of the peak heights, corresponding to 2αm

I(x,y) = I0(x,y)+∑

m

2αm

ir(x,y)im(x,y) |cmr(τ −τmr)| cos(ψ).

(11)

?ir(x,y)·im(x,y), is

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then given. A correction by the reference arm intensity (which was independently measured)

finally gives the intensity profiles of the modes.

We quantify modal discrimination with the multi-path interference (MPI) value.

It describes the ratio of modal intensity of a higher order mode to the modal

intensity of the fundamental LP01

mode. Specifically, it is defined as MPI =

10 log10(α2

m

? ?im(x,y)dxdy/? ?iLP01(x,y)dxdy).

Resolving small intermodal group-delays 4.

To reconstruct the modes of LMA fibers, it is required to temporally resolve their small relative

intermodal group-delays. According to Eq. (11), the temporal resolution of C2imaging is re-

lated to the width of the cross-correlation integral cmr, which in turn is dependent on dispersion

and source bandwidth. Particularly, for a parabolic dispersion mismatch in the interferometer,

Eq. (10) may be also written as c(t) = (1/2π)?dΩ S(Ω) exp(i(Δϕ(2)Ω2/2)) exp(−iΩ·t).

FWHM of the absolute value of c(t) ) can be calculated as a function of the bandwidth of the

spectrum and the residual group-velocity dispersion (GVD) Δϕ(2).

For a given shape of the spectrum, the temporal resolution ΔτFWHM (defined here as the

05 1015 20

0

1

2

3

4

5

Bandwidth ΔλFWHM [nm]

(a)

Temporal Resolution

ΔτFWHM [ps]

numerical solution

coherence time

dispersive broadening

05 101520

Bandwidth ΔλFWHM [nm]

(b)

0

1

2

3

4

>5

Temporal Resolution

ΔτFWHM [ps]

0

0.2

0.4

0.6

0.8

1.0

GVD [ps ]

2

Fig. 2. (a) Temporal resolution as a function of the FWHM spectral bandwidth of a Gaus-

sian spectrum (for GVD value of φ(2)= 0.1ps2), (b) and as a function of both FWHM

bandwidth and GVD.

For a Gaussian spectrum at 1060 nm and residual GVD of Δϕ(2)= 0.1ps2(corresponding

to about 4m of LMA fiber and no dispersion compensating fiber in the reference arm), Fig.

2(a) shows the temporal resolution as a function of bandwidth. For small bandwidths, disper-

sion plays a minor role, and the resolution is governed by the coherence time, which can be

defined as Δτcoher

gular frequency (ΔωFWHM≈ 2πc0·ΔλFWHM/λ2

may indicate the possibility of an excellent temporal resolution. However, if the interferometer

is not dispersion balanced, then the cross-correlation broadens for larger spectral bandwidths

and the resolution is governed by dispersive broadening: Δτdisp

consequence, to obtain a good temporal resolution in the presence of residual dispersion, an

optimal bandwidth of the spectrum (i.e. appropriate choice of bandpass) must be found.

FWHM= 8·ln(2)/ΔωFWHM. Where ΔωFWHMis the width of the spectrum in an-

0). In the absence of dispersion, broad spectra

FWHM= Δϕ(2)·ΔωFWHM. As a

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Figure 2(b) shows the temporal resolution as a function of both bandwidth and GVD for a

Gaussian spectrum. It can be seen that for increasing GVD the optimum bandwidth gets slightly

shorter. The horizontal line in Fig. 2(b) highlights the parameter configuration shown in Fig.

2(a).

To obtain the best resolution even for broad bandwidths, the dispersion of the two arms

of the interferometer must be matched (see also Eq. (10)). This technique is well-known in

optical coherence tomography (OCT) [14]. In our setup, we insert a single mode fiber in the

reference arm which balances the dispersion of the test fiber. Dispersion can only be exactly

matched for one mode. For the other modes, residual dispersive phases remain which cause

broadening of their peaks in the cross-correlation trace. So, dispersion balancing requires test

fibers whose modes show similar dispersion values. This condition is fulfilled for most LMA

fibers containing a few modes. Then, the temporal resolution is given by the coherence time.

For sufficiently broad spectra, intermodal delays of a few femtoseconds can be resolved.

The shape and smoothness of the spectrum also have an impact on the cross-correlation trace,

and thus, the temporal resolution. These influences have been discussed in the context of OCT,

e. g. [15]. In this paper we will consider the impact of one the two spectrum parameters, namely

spectral shape (as demonstrated in section 6).

5.Comparison with other beam characterization methods

Laser-beam quality is often measured in terms of the M2-parameter [16]. This quantifier has

also been applied for the beam characterization of fiber lasers. However, there exist situations

in which the M2-parameter is low (indicating good beam quality), yet the beam contains a

significant amount of higher-order mode content [17]. The resulting modal interference causes

beam fluctuations in the far-field - an apparent sign of an unstable, and thus, bad beam output.

A recently reported alternative characterization technique - S2imaging - enables the direct

measurementofmodalcontent.Essentially,inthismethod(inwhichanexternalreferencebeam

is not present) the offset term of the total intensity, I0(x,y) of Eq. (6), is spectrally resolved

[8]. However, the interference between each mode with every other mode, as described by

the last term in Eq. (6), complicates the analysis. Specifically, one of the modes, usually the

fundamental mode, should be used as a ’reference’ mode for the analysis of the interference

term. In this case, a reasonable estimate of the MPI values of the modes may be obtained only

when this mode has the largest power in the propagating beam. Consequently, the method fails

in the general case of multiple modes having similar power levels.

Our method of cross-correlation (C2) imaging employs the external reference beam, and

therefore, is not limited to the case of one dominating mode and few weaker ones. The ref-

erence beam also offers a new degree of freedom, for example, to reveal polarization of the

waveguide modes. In contrast to S2imaging, all the modes having arbitrary relative power lev-

els are measured independently of one another. However, to obtain reliable measures of the

corresponding MPI values, one needs to ensure flatness of the offset intensity I0as a function

of delay-stage position. This condition is not present in S2imaging in which all the modes

propagate in the same fiber, and thus, experience similar intensity fluctuations.

As we will show in the following sections, we can obtain modal discrimination approaching

- 30 dB, indicating that the flatness of I0is not a debilitating problem for C2imaging. Future

studies will reveal the exact origin of noise in I0, which would help achieving modal discrim-

ination values even better than -30 dB currently achievable by our C2imaging technique. The

impact of the image acquisition on modal discrimination is also worth noting: noise of the cam-

era plays a role but also discretization, for instance, we use a 16-bit camera, corresponding to

MPIs up to 10log10(2−15)= - 45 dB (note, that half of the range is needed for the maximum I0

intensity).

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6.Experimental results

We will now demonstrate the versatility of the C2imaging using a specialty higher-order mode

(HOM) fiber, as well as a LMA-fiber that is usually employed in high-power fiber lasers. The

HOM-fiber possesses modes exhibiting distinct dispersive behavior. This allows us to illus-

trate the impact of dispersion on the modal content. To demonstrate the concept of dispersion

compensation, we employ an LMA-fiber that contains modes that show similar dispersive char-

acteristics. This permits high temporal resolution of small intermodal group delays.

6.1. Specialty fiber with modes of distinct dispersion

To reveal the impact of dispersion on the modal content, we use a test fiber in which the modes

show different dispersion values. For this it is preferable to use a well know dispersion char-

acteristic in the reference arm. However, to obtain the following results, we used almost only

free-space path in the reference arm. A schematic of the setup is shown in Fig. 1. The few-mode

fiber under test is the final element of a module (L=0.6 m) consisting of a single-mode fiber,

a turn around point long-period grating (TAP LPG), and the higher-order mode (HOM) fiber

(L=0.4 m) [13,18]. In this fiber, the LPG spectrum characterizes the mode conversion efficiency

from the LP01core-mode to the LP02core-mode. The measured LPG conversion will be used

as an independent reference for comparison against the ratio of modal weights obtained by C2

imaging.

(b)

(c)

56789 10

0

0.5

1

Delay [ps]

012345

0

0.5

1

Delay [ps]

Cross-correlation

signal [a.u.]

envelope

model fit

envelope

model fit

(a)

Delay [ps]

Cross-correlation

signal [a.u.]

02468 10

−3

−2

−1

0

1

2

3

experimental data

envelope

LP

LP

01

02

LP01

LP02

Cross-correlation

signal [a.u.]

Fig. 3. (a) Cross-correlation trace for the entire image (data is offset corrected) for the

bandpass at λcenter=780 nm. (b) and (c), fit of the model to the envelope of the experimental

data, for the first and second peak, corresponding to LP01and LP02, respectively.

Figure 3(a) shows an example of the total cross-correlation between the reference field and

the output of the fiber (integrated over all pixels of the camera). The peaks in the trace cor-

respond to the two different modes in the HOM fiber. We analyze the data using Eq. (11). In

a first step, we detect the envelope of this signal, which is shown as a red line in Fig. 3(a).

In Fig. 3(b) and (c) the extracted envelope around the two peaks is shown together with a fit

of |c(τ)| onto this data. To apply Eq. (10), the spectrum (behind the bandpass of the refer-

ence arm only) must be measured. Since the bandpass only has a spectral width of 4 nm, the

wavelength-dependent LPG spectrum has a negligible impact on the spectra of the two modes

(the LPG mode-conversion bandwidth exceeds 20 nm, in this case). The fitting is based on

the same spectrum. The difference in shape of the envelopes for the two modes is due to the

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770 780790800

−600

−400

−200

0

200

Wavelength [nm]

(b)

Dispersion [ps/nm/km]

exp. LP01

exp. LP02

Theory

770780 790 800

0

5

10

15

20

Wavelength [nm]

(a)

Group-delay (+offset) [ps/m]

exp. LP01

exp. LP02

Theory

Fig. 4. (a) Group-delays, and (b) Dispersion values of the two modes as a function of center

wavelength of the bandpass.

770780790800

−25

−20

−15

−10

−5

0

Wavelength [nm]

(c)

MPI [dB]

experimental data

from LPG loss

(a) (b)

LP

LP01

02

Fig. 5. (a) and (b), reconstructed LP01and LP02-mode (gamma-adjusted) at a center wave-

length of 780 nm, (c) multi-path interference (MPI) values as a function of center wave-

length of the bandpass.

impact of dispersion. The side-lobes around the dominant peaks (clearly visible for the LP02

mode) are due to the steep edges of the bandpass filter (to avoid this ringing, and thus, to obtain

a better temporal resolution, it would be advantageous to use a spectrum that does not show

these features). By fitting |c(τ)| on the data, the relative group-delay and the dispersion are

found for every mode. Furthermore, the group-delay and dispersion can be retrieved as a func-

tion of wavelength by shifting the center wavelength of the filtered spectrum via tilting of the

4-nm bandpass. Figures 4(a) and (b) show the dispersion and relative group-delays as a func-

tion of wavelength, respectively. The data points are compared to the output of a mode-solver

simulating the fiber under test. It is worth noting that the experimental group-delay data refers

to a common reference mark, which is provided by the backlash correction of the motorized

translation stage.

Figures 5(a) and (b) show the retrieved LP01and LP02modes, respectively. Moreover the

relative (dispersion-corrected) weights of the normalized modes can be obtained. To demon-

strate the accuracy of the C2imaging, in Fig. 5(c), we show the multi-path interference (MPI)

value as a function of wavelength. The obtained MPI values match very well with the mode

conversion efficiencies independently measured by recording the LPG spectrum.

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6.2.Large-mode area fibers

For characterization of large-mode area (LMA) fibers, in which all the modes have similar

magnitudes of chromatic dispersion and the relative delays are on a picosecond (or less)

timescales, the dispersion matching scheme, which has been described in section 4, can

be employed. The resulting better temporal resolution reveals modes at small intermodal

group-delays. The polarization maintaining LMA fiber under test has a core diameter of

approximately 27.5 μm and a NA of 0.062. We use a polarizer and a pair of half-wave plates

to ensure launch of the beam into one of the dominant polarization axes. The fiber is 5 m long.

The dispersion is matched by inserting 4.08 m of single-moded HI-1060 fiber in the reference

arm. This length has been determined by cutting the input of the reference fiber until the width

of the dominant peak in the cross-correlation matches the one calculated from the measured

(full) spectrum.

1000 1025 1050 1075 1100 1125

Wavelength [nm]

(a)

0

0.2

0.4

0.6

0.8

1

Intensity [a.u.]

w/o bandpass

with bandpass

−2 −101

Delay [ps]

234567

0

0.1

0.2

0.3

0.4

w/o bandpass

with bandpass

(b)

Cross-correlation

signal [a.u.]

LP

21

LP01

LP

11

Fig. 6. (a) Spectrum without and after filtering with the 5-nm bandpass, (b) corresponding

envelopes of the cross-correlation traces.

MPI = -14.6 dBMPI = -16.6 dBMPI = - 29 dBMPI = - 19 dB

Fig. 7. Reconstructed mode profiles in order of temporal delay as shown in the cross-

correlation trace of Fig. 6(b) for the case of the full spectrum.

If the dispersion is matched, a broad spectrum results in a better temporal resolution. To

demonstrate this effect, we record cross-correlation traces with the full spectrum, as well as

with a spectrum that was filtered with a 5-nm bandpass filter. Figure 6(a) shows the full and

filtered spectrum. The corresponding cross-correlation traces (for the entire image) are shown

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(a)

(b)

(c)

−2 −101234567

0

0.1

0.2

0.3

0.4

Delay in [ps]

Cross-correlation

signal [a.u.]

Excitation (a)

Excitation (b)

Excitation (c)

Fig. 8. (a-c) Output of the fiber under test (near-field images) for different excitations, and

corresponding changes in the cross-correlation trace.

in Fig. 6(b). It can be seen that by using the full spectrum, the temporal resolution signifi-

cantly improves to values smaller than 300 fs, and as a consequence, the odd and even modes

for the LP11and LP21modes can be resolved. The impact of the shape of the spectrum on the

cross-correlation trace is also revealed: Since the filtered spectrum shows steep edges, the corre-

sponding cross-correlation trace shows ringing (especially around the peak of the LP01-mode).

In contrast, the full spectrum is bell-shaped, which causes a cleaner cross-correlation trace.

Figure 7 displays all the reconstructed modes in order of their delay shown in Fig. 6(b). The

(dispersion corrected) MPI values for the reconstructed modes corresponding to Fig. 6(b) (full

spectrum) are -14.6 dB and -16.6 dB for the two LP11modes, and -29 dB and -19 dB for the

two LP21modes. Note that the cross-correlation signal shown in Fig. 6(b) is proportional to

the square-root of the modal intensity, as described by Eq. (11). It is worth noting that for a V

parameter around 5, the next higher LP02-mode should be also present, however, it could not

be observed. This is mainly attributed to the fact that the coiling of the fiber to a diameter of

around 30 cm probably stripped off the LP02-mode.

A qualitative observation based upon the image of the output of the fiber, as shown in Fig.

8(a) when the reference beam is blocked, suggests fundamental-mode operation. In contrast, the

mode-measurementtechniquequantitativelyrevealsthatthebeamcontainsasignificantamount

of power in the higher-order modes. The versatility of C2imaging is illustrated by changing the

launching conditions into the fiber. As a consequence, we demonstrate the subsequent change

in modal content distribution. This is displayed in Fig. 8. For the example, the fiber-output

as shown in Fig. 8(b) corresponds to a stronger excitation of the ’slow’ LP11-mode, while the

fiber-output, as shown in Fig. 8(c), results in a stronger ’fast’ LP11-mode. However, in spite of

the distorted output beam in these two situations, LP01-mode still carries the most modal power.

This proves that C2imaging does not rely on a dominant fundamental mode.

Also note that the temporal splitting between the LP21modes is more pronounced as com-

pared to the delay between the LP11modes. Thus, modes with higher orbital angular momentum

(i.e. LPlmmodes with higher l) become more susceptible to the birefringence of this polariza-

tion maintaining fiber (in accord with [19]). Thus, C2imaging quantitatively determines the

change in power distribution of the modes for the different excitations.

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7.Summary and conclusions

In conclusion, we have presented versatile mode-measurement using C2imaging in which a

reference beam samples all the modes of the fiber under test. In this way, we demonstrate reli-

able reconstruction of the mode content without making any assumption about the fiber under

test. By analyzing the slowly varying envelope of the interferometric output, we show, for the

first time, to the best of our knowledge, the impact of dispersion on the modal content. This, in

turn, allows us to use C2imaging for MPI and dispersion measurement hitherto not attempted

with optical low-coherence interferometry. In a specialty higher-order mode fiber, this allowed

accurate retrieval of modes that have small relative group delays but distinct dispersive behav-

ior. At an optimum spectral bandwidth of 4 nm, we measure intermodal delays as short as 2.8 ps

with mode-extinction values approaching -20 dB. The reconstructed modal weights agree with

the mode-conversion of the long-period grating spectrum. With polarization maintaining LMA

fibers we demonstrate the concept of dispersion matching in the interferometer, this permits

a temporal resolution smaller than 300 fs, which is sufficient to even reveal temporal splitting

between odd and even higher order modes due to the impact of birefringence of the polarization

maintaining fiber. C2imaging also permits studying the modal content distribution for differ-

ent launching conditions into the LMA-fiber. MPI values down to - 30 dB can be accurately

retrieved. The versatility of C2imaging makes the it highly attractive for the characterization

of planar waveguide devices which may have undesired (slab) modes, as well as fibers that are

being developed for next-generation high-power fiber lasers.

Appendix

In the section discussing the data-analysis, we derived the general expression, Eq. (11), by

assuming that the phase of the cross-correlation integral, Eq. (10), is slowly varying. For the

example of a Gaussian spectrum S(Ω) = S0·exp(−(Ω/ΔΩ)2), Im(x,y,τ) will be given by

Im(x,y,τ) = αm|em(x,y)·er(x,y)|S0ΔΩ

(1+d2

√π

1

m)1/4exp

?

−(τ −τmr)2ΔΩ2

4(1+d2

m)

?

cos(ψ),

(12)

where dmstands for (β(2)

m ·L−β(2)

ψ = φm(x,y)+Θmr+(τ −τmr)2ΔΩ2

r

·Lr)ΔΩ2/2, the phase is given by

4(1+d2

m)

dm+const,

(13)

where φm(x,y) is the spatial phase of the mode em(x,y). As already defined in the data-analysis

section, Θmr=

r

·Lr−τ ·ω0

spectral width of a few THz, the third term in Eq. (13) will be negligible compared to τ ·ω0in

Θmr. Thus, a separation of the envelope and fast oscillation is possible.

Moreover, it can be seen that if τ = τmr then the term 1/(1+d2

determines the peak-height of the envelope in the cross-correlation signal. Particularly, if the

modes of the test fiber show distinct dispersive behavior then this term will be relevant for the

correct determination of the modal weights.

?

β(0)

m ·L−β(0)

?

, and τmr= (L/vgr,m−Lr/vgr,r). Particularly, for

m)1/4will still remain and

Acknowledgments

R. A. Barankov and D. N. Schimpf have contributed equally to the theoretical and experi-

mental realization of C2imaging. The authors thank K. Jespersen from OFS Fitel Denmark

for providing the HOM-fiber and the TAP-LPG, and B. Samson from Nufern for providing the

polarization maintaining LMA fiber. This work was partly funded by ARL Grant No. W911NF-

06-2-0040.

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