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.

© 2011 OSA

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 [37], 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.

A promising technique called spatially and spectrally resolved (S2) imaging has been recently 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 characteristics 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 C2 imaging. 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]. C2 imaging 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.

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.

 

Fig. 1 Schematic of the experimental setup (SLD: superluminescent diode), and illustration of the cross-correlation trace expected at one pixel of the stack of images.

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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 modifications to a wide variety of fibers ranging from LMA fibers, with small intermodal group-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 develop 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. S2 imaging).

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

I(x,y)=ΔT/2+ΔT/2dt|E(x,y,t)|2=+dω2π|E(x,y,ω)|2,
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 representation in frequency domain.

The electric field in frequency domain, E(ω) = ∫ dte iωt E(t), is given by a superposition of the field coming out of the fiber and the reference field:

E(x,y,ω)=mαmem(x,y,ω)Am(ω)eiϕm+er(x,y,ω)Ar(ω)eiϕr.
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 e m (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
ϕm=βm(ω)L.
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+ωcd,
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 Lr exhibiting 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 e r (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 I 0 and the term Iint, which is due to interference between the reference field and the individual modes:

I(x,y)=I0(x,y)+Iint(x,y),
with the two terms written explicitly as
I0=+dω2π(|erAr(ω)|2+m|αmemAm(ω)|2+(mm)2Re[αmem*Am*(ω)αmemAm(ω)ei(ϕmϕm)])
Iint=m+dω2π2Re[er*(x,y,ω)Ar*(ω)αmem(x,y,ω)Am(ω)ei(ϕmϕr)].

The term I 0(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 convenient 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,ω 0). Furthermore, the phase-difference Δϕmr = (ϕmϕr) in Eq. (7) is Taylor-expanded around the center angular frequency ω 0 in terms of the angular frequency difference Ω = ωω 0 as follows

Δϕmr=Θmr(ττmr)Ω+Δϕmr(Ω),
where the phase-mismatch between the two arms is Θmr=(βm(0)Lβr(0)Lrτω0), a delay variable τ is defined as d/c, τmr stands for a group-delay difference τmr = (L/vgr,mLr /vgr,r), and the dispersion mismatch between the two arms of the interferometer is described by Δϕmr=Σk2(βm(k)Lβr(k)Lr)Ωk/k!, where the β ( k ) stand for the Taylor-coefficients of the mode-propagation constant β.

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

Iint(x,y)=mIm(x,y,τ)=m2αmRe[er*(x,y,ω0)em(x,y,ω0)cmr(ττmr)exp(iΘmr)],
where the cross-correlation function cmr is given by
cmr(t)=12πdΩS(Ω)exp(iΔϕmr(Ω))exp(iΩt).
It describes the impact of the frequency-dependent mode-propagation constant on the signal. Particularly, the group-delay τmr determines the position of m-th mode in the signal (as illustrated 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 oscillating 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:

I(x,y)=I0(x,y)+m2αmir(x,y)im(x,y)|cmr(ττmr)|cos(ψ).
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 (ττmr) |. 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 I 0(x, y) only. The resulting term |I(x,y) – I 0(x, y)| is integrated over all (x,y) pixels. This allows us to obtain a one-dimensional signal as a function of the translation stage position from which we can determine the envelope of the 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 information about the dispersion of the modes: The fitting of |cmr (ττmr) |, using the measured 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 2αmir(x,y)im(x,y), is 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 LP 01 mode. Specifically, it is defined as MPI=10log10(αm2im(x,y)dxdy/iLP01(x,y)dxdy).

4. Resolving small intermodal group-delays

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 C2 imaging is related 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). For a given shape of the spectrum, the temporal resolution ΔτFWHM (defined here as the 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 Gaussian spectrum at 1060 nm and residual GVD of Δφ(2) = 0.1ps 2 (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, dispersion plays a minor role, and the resolution is governed by the coherence time, which can be defined as ΔτFWHMcoher=8ln(2)/ΔωFWHM. Where ΔωFWHM is the width of the spectrum in angular frequency ( ΔωFWHM2πc0ΔλFWHM/λ02). In the absence of dispersion, broad spectra 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: ΔτFWHMdisp=Δφ(2). As a 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.

 

Fig. 2 (a) Temporal resolution as a function of the FWHM spectral bandwidth of a Gaussian spectrum (for GVD value of ϕ (2) = 0.1ps 2), (b) and as a function of both FWHM bandwidth and GVD.

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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 - S2 imaging - enables the direct measurement of modal content. Essentially, in this method (in which an external reference beam is not present) the offset term of the total intensity, I 0(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 reference beam also offers a new degree of freedom, for example, to reveal polarization of the waveguide modes. In contrast to S2 imaging, all the modes having arbitrary relative power levels 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 I 0 as a function of delay-stage position. This condition is not present in S2 imaging 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 I 0 is not a debilitating problem for C2 imaging. Future studies will reveal the exact origin of noise in I 0, which would help achieving modal discrimination values even better than −30 dB currently achievable by our C2 imaging technique. The impact of the image acquisition on modal discrimination is also worth noting: noise of the camera 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 I 0 intensity).

6. Experimental results

We will now demonstrate the versatility of the C2 imaging 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 illustrate 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 characteristics. 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 characteristic 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 LP01 core-mode to the LP02 core-mode. The measured LPG conversion will be used as an independent reference for comparison against the ratio of modal weights obtained by C2 imaging.

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 correspond 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 reference 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 impact of dispersion. The side-lobes around the dominant peaks (clearly visible for the LP 02 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 function 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 function 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.

 

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 LP01 and LP02, respectively.

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

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Figures 5(a) and (b) show the retrieved LP 01 and LP 02 modes, respectively. Moreover the relative (dispersion-corrected) weights of the normalized modes can be obtained. To demonstrate the accuracy of the C2 imaging, 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.

 

Fig. 5 (a) and (b), reconstructed LP 01 and LP 02-mode (gamma-adjusted) at a center wavelength of 780 nm, (c) multi-path interference (MPI) values as a function of center wavelength of the bandpass.

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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.

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 in Fig. 6(b). It can be seen that by using the full spectrum, the temporal resolution significantly improves to values smaller than 300 fs, and as a consequence, the odd and even modes for the LP 11 and LP 21 modes 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 corresponding cross-correlation trace shows ringing (especially around the peak of the LP 01-mode). In contrast, the full spectrum is bell-shaped, which causes a cleaner cross-correlation trace.

 

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

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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 LP 11 modes, and −29 dB and −19 dB for the two LP 21 modes. 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 LP 02-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 LP 02-mode.

 

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.

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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-measurement technique quantitatively reveals that the beam contains a significant amount of power in the higher-order modes. The versatility of C2 imaging 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’ LP 11-mode, while the fiber-output, as shown in Fig. 8(c), results in a stronger ’fast’ LP 11-mode. However, in spite of the distorted output beam in these two situations, LP 01-mode still carries the most modal power. This proves that C2 imaging does not rely on a dominant fundamental mode.

 

Fig. 8 (a–c) Output of the fiber under test (near-field images) for different excitations, and corresponding changes in the cross-correlation trace.

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Also note that the temporal splitting between the LP 21 modes is more pronounced as compared to the delay between the LP 11 modes. Thus, modes with higher orbital angular momentum (i.e. LPlm modes with higher l) become more susceptible to the birefringence of this polarization maintaining fiber (in accord with [19]). Thus, C2 imaging quantitatively determines the change in power distribution of the modes for the different excitations.

7. Summary and conclusions

In conclusion, we have presented versatile mode-measurement using C2 imaging in which a reference beam samples all the modes of the fiber under test. In this way, we demonstrate reliable 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 C2 imaging 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 behavior. 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. C2 imaging also permits studying the modal content distribution for different launching conditions into the LMA-fiber. MPI values down to – 30 dB can be accurately retrieved. The versatility of C2 imaging 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) = S 0 · exp(–(Ω/ΔΩ)2), Im(x, y, τ) will be given by

Im(x,y,τ)=αm|em(x,y)er(x,y)|S0ΔΩπ1(1+dm2)1/4exp[(ττmr)2ΔΩ24(1+dm2)]cos(ψ),
where dm stands for (βm(2)Lβr(2)Lr)ΔΩ2/2, the phase is given by
ψ=ϕm(x,y)+Θmr+(ττmr)2ΔΩ24(1+dm2)dm+const,
where ϕm(x, y) is the spatial phase of the mode e m(x, y). As already defined in the data-analysis section, Θmr=(βm(0)Lβr(0)Lrτω0), and τmr = (L/vgr,mLr /vgr,r). Particularly, for spectral width of a few THz, the third term in Eq. (13) will be negligible compared to τ · ω 0 in 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+dm2)1/4 will still remain and 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.

Acknowledgments

R. A. Barankov and D. N. Schimpf have contributed equally to the theoretical and experimental realization of C2 imaging. 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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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. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Express 36, 689–691 (2011). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-5-689

7. J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25, 442–444 (2000). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-25-7-442 [CrossRef]  

8. J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron. 15, 61–70 (2009). [CrossRef]  

9. S. Blin, D. M. Nguyen, T. N. Nguyen, M. Thual, T. Chartier, and L. Provino, “Simple modal analysis method for multi-mode fibers,” European Conference on Optical Communication (ECOC) (2009), P1.16.

10. Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry, ” Appl. Phys. B 96, 345–353 (2009). [CrossRef]  

11. P. Nandi, Z. Chen, A. Witkowska, W. J. Wadsworth, T. A. Birks, and J. C. Knight, “Characterization of a photonic crystal fiber mode converter using low coherence interferometry,” Opt. Lett. 34, 1123–1125 (2009). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-7-1123 [CrossRef]   [PubMed]  

12. P. Hamel, Y. Jaoun, R. Gabet, and S. Ramachandran, “Optical low-coherence reflectometry for complete chromatic dispersion characterization of few-mode fibers,” Opt. Lett. 32, 1029–1031 (2007). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-32-9-1029 [CrossRef]   [PubMed]  

13. S. Ramachandran, “Dispersion-tailored few-mode fibers: a versatile platform for in-fiber photonic devices,” J. Lightwave Technol. 23, 3426–3443 (2005). [CrossRef]  

14. M. Wojtkowski, V. Srinivasan, T. Ko, J. Fujimoto, A. Kowalczyk, and J. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12, 2404–2422 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-11-2404 [CrossRef]   [PubMed]  

15. J. F. de Boer, C. E. Saxer, and J. S. Nelson, “Stable carrier generation and phase-resolved digital data processing in optical coherence tomography,” Appl. Opt. 40, 5787–5790 (2001). http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-40-31-5787. [CrossRef]  

16. X. Luo, P. Chen, and Y. Wang, “Power content M2-values smaller than one,” Appl. Phys. B 98, 181185 (2010).

17. S. Wielandy, “Implications of higher-order mode content in large mode area fibers with good beam quality,” Opt. Express 15, 15402–15409 (2007). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-23-15402 [CrossRef]   [PubMed]  

18. K. G. Jespersen, T. Le, L. Grner-Nielsen, D. Jakobsen, M. E. V. Pederesen, M. B. Smedemand, S. R. Keiding, and B. Palsdottir, “A higher-order-mode fiber delivery for Ti:sapphire femtosecond lasers,” Opt. Express 18, 7798–7806 (2010). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-7798 [CrossRef]   [PubMed]  

19. S. Golowich and S. Ramachandran, “Impact of fiber design on polarization dependence in microbend gratings,” Opt. Express 13, 6870–6877 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-18-6870 [CrossRef]   [PubMed]  

References

  • View by:
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  • |

  1. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B 27, B63–B92 (2010). http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-27-11-B63
    [CrossRef]
  2. D. N. Schimpf, J. Limpert, and A. Tünnermann, “Optimization of high performance ultra-fast fiber laser systems to >10GW peak power,” J. Opt. Soc. Am. B 27, 2051–2060 (2010). http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-27-10-2051
    [CrossRef]
  3. L. Dong, H. A. Mckay, A. Marcinkevicius, L. Fu, J. Li, B. K. Thomas, and M. E. Fermann, “Extending effective area of fundamental mode in optical fibers,” J. Lightwave Technol. 27, 1565–1570 (2009). http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-27-11-1565
    [CrossRef]
  4. S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultra-large modal areas in optical fibers,” Opt. Lett. 27, 1797–1799 (2006). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-31-12-1797
    [CrossRef]
  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. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Express 36, 689–691 (2011). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-5-689
  7. J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25, 442–444 (2000). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-25-7-442
    [CrossRef]
  8. J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron. 15, 61–70 (2009).
    [CrossRef]
  9. S. Blin, D. M. Nguyen, T. N. Nguyen, M. Thual, T. Chartier, and L. Provino, “Simple modal analysis method for multi-mode fibers,” European Conference on Optical Communication (ECOC) (2009), P1.16.
  10. Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry, ” Appl. Phys. B 96, 345–353 (2009).
    [CrossRef]
  11. P. Nandi, Z. Chen, A. Witkowska, W. J. Wadsworth, T. A. Birks, and J. C. Knight, “Characterization of a photonic crystal fiber mode converter using low coherence interferometry,” Opt. Lett. 34, 1123–1125 (2009). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-7-1123
    [CrossRef] [PubMed]
  12. P. Hamel, Y. Jaoun, R. Gabet, and S. Ramachandran, “Optical low-coherence reflectometry for complete chromatic dispersion characterization of few-mode fibers,” Opt. Lett. 32, 1029–1031 (2007). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-32-9-1029
    [CrossRef] [PubMed]
  13. S. Ramachandran, “Dispersion-tailored few-mode fibers: a versatile platform for in-fiber photonic devices,” J. Lightwave Technol. 23, 3426–3443 (2005).
    [CrossRef]
  14. M. Wojtkowski, V. Srinivasan, T. Ko, J. Fujimoto, A. Kowalczyk, and J. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12, 2404–2422 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-11-2404
    [CrossRef] [PubMed]
  15. J. F. de Boer, C. E. Saxer, and J. S. Nelson, “Stable carrier generation and phase-resolved digital data processing in optical coherence tomography,” Appl. Opt. 40, 5787–5790 (2001). http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-40-31-5787 .
    [CrossRef]
  16. X. Luo, P. Chen, and Y. Wang, “Power content M2-values smaller than one,” Appl. Phys. B 98, 181185 (2010).
  17. S. Wielandy, “Implications of higher-order mode content in large mode area fibers with good beam quality,” Opt. Express 15, 15402–15409 (2007). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-23-15402
    [CrossRef] [PubMed]
  18. K. G. Jespersen, T. Le, L. Grner-Nielsen, D. Jakobsen, M. E. V. Pederesen, M. B. Smedemand, S. R. Keiding, and B. Palsdottir, “A higher-order-mode fiber delivery for Ti:sapphire femtosecond lasers,” Opt. Express 18, 7798–7806 (2010). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-7798
    [CrossRef] [PubMed]
  19. S. Golowich and S. Ramachandran, “Impact of fiber design on polarization dependence in microbend gratings,” Opt. Express 13, 6870–6877 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-18-6870
    [CrossRef] [PubMed]

2009

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron. 15, 61–70 (2009).
[CrossRef]

Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry, ” Appl. Phys. B 96, 345–353 (2009).
[CrossRef]

2005

1811

X. Luo, P. Chen, and Y. Wang, “Power content M2-values smaller than one,” Appl. Phys. B 98, 181185 (2010).

Chen, P.

X. Luo, P. Chen, and Y. Wang, “Power content M2-values smaller than one,” Appl. Phys. B 98, 181185 (2010).

Fini, J. M.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron. 15, 61–70 (2009).
[CrossRef]

Leuchs, G.

Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry, ” Appl. Phys. B 96, 345–353 (2009).
[CrossRef]

Luo, X.

X. Luo, P. Chen, and Y. Wang, “Power content M2-values smaller than one,” Appl. Phys. B 98, 181185 (2010).

Ma, Y. Z.

Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry, ” Appl. Phys. B 96, 345–353 (2009).
[CrossRef]

Mermelstein, M. D.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron. 15, 61–70 (2009).
[CrossRef]

Nicholson, J. W.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron. 15, 61–70 (2009).
[CrossRef]

Onishchukov, G.

Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry, ” Appl. Phys. B 96, 345–353 (2009).
[CrossRef]

Peschel, U.

Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry, ” Appl. Phys. B 96, 345–353 (2009).
[CrossRef]

Ramachandran, S.

Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry, ” Appl. Phys. B 96, 345–353 (2009).
[CrossRef]

S. Ramachandran, “Dispersion-tailored few-mode fibers: a versatile platform for in-fiber photonic devices,” J. Lightwave Technol. 23, 3426–3443 (2005).
[CrossRef]

Schmauss, B.

Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry, ” Appl. Phys. B 96, 345–353 (2009).
[CrossRef]

Sych, Y.

Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry, ” Appl. Phys. B 96, 345–353 (2009).
[CrossRef]

Wang, Y.

X. Luo, P. Chen, and Y. Wang, “Power content M2-values smaller than one,” Appl. Phys. B 98, 181185 (2010).

Yablon, A. D.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron. 15, 61–70 (2009).
[CrossRef]

Appl. Phys. B

Y. Z. Ma, Y. Sych, G. Onishchukov, S. Ramachandran, U. Peschel, B. Schmauss, and G. Leuchs, “Fiber-modes and fiber-anisotropy characterization using low-coherence interferometry, ” Appl. Phys. B 96, 345–353 (2009).
[CrossRef]

X. Luo, P. Chen, and Y. Wang, “Power content M2-values smaller than one,” Appl. Phys. B 98, 181185 (2010).

IEEE J. Sel. Top. Quantum Electron.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron. 15, 61–70 (2009).
[CrossRef]

J. Lightwave Technol.

Other

S. Blin, D. M. Nguyen, T. N. Nguyen, M. Thual, T. Chartier, and L. Provino, “Simple modal analysis method for multi-mode fibers,” European Conference on Optical Communication (ECOC) (2009), P1.16.

S. Wielandy, “Implications of higher-order mode content in large mode area fibers with good beam quality,” Opt. Express 15, 15402–15409 (2007). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-23-15402
[CrossRef] [PubMed]

K. G. Jespersen, T. Le, L. Grner-Nielsen, D. Jakobsen, M. E. V. Pederesen, M. B. Smedemand, S. R. Keiding, and B. Palsdottir, “A higher-order-mode fiber delivery for Ti:sapphire femtosecond lasers,” Opt. Express 18, 7798–7806 (2010). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-7798
[CrossRef] [PubMed]

S. Golowich and S. Ramachandran, “Impact of fiber design on polarization dependence in microbend gratings,” Opt. Express 13, 6870–6877 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-18-6870
[CrossRef] [PubMed]

P. Nandi, Z. Chen, A. Witkowska, W. J. Wadsworth, T. A. Birks, and J. C. Knight, “Characterization of a photonic crystal fiber mode converter using low coherence interferometry,” Opt. Lett. 34, 1123–1125 (2009). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-7-1123
[CrossRef] [PubMed]

P. Hamel, Y. Jaoun, R. Gabet, and S. Ramachandran, “Optical low-coherence reflectometry for complete chromatic dispersion characterization of few-mode fibers,” Opt. Lett. 32, 1029–1031 (2007). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-32-9-1029
[CrossRef] [PubMed]

M. Wojtkowski, V. Srinivasan, T. Ko, J. Fujimoto, A. Kowalczyk, and J. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12, 2404–2422 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-11-2404
[CrossRef] [PubMed]

J. F. de Boer, C. E. Saxer, and J. S. Nelson, “Stable carrier generation and phase-resolved digital data processing in optical coherence tomography,” Appl. Opt. 40, 5787–5790 (2001). http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-40-31-5787 .
[CrossRef]

D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B 27, B63–B92 (2010). http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-27-11-B63
[CrossRef]

D. N. Schimpf, J. Limpert, and A. Tünnermann, “Optimization of high performance ultra-fast fiber laser systems to >10GW peak power,” J. Opt. Soc. Am. B 27, 2051–2060 (2010). http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-27-10-2051
[CrossRef]

L. Dong, H. A. Mckay, A. Marcinkevicius, L. Fu, J. Li, B. K. Thomas, and M. E. Fermann, “Extending effective area of fundamental mode in optical fibers,” J. Lightwave Technol. 27, 1565–1570 (2009). http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-27-11-1565
[CrossRef]

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultra-large modal areas in optical fibers,” Opt. Lett. 27, 1797–1799 (2006). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-31-12-1797
[CrossRef]

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).

F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Express 36, 689–691 (2011). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-5-689

J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25, 442–444 (2000). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-25-7-442
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic of the experimental setup (SLD: superluminescent diode), and illustration of the cross-correlation trace expected at one pixel of the stack of images.

Fig. 2
Fig. 2

(a) Temporal resolution as a function of the FWHM spectral bandwidth of a Gaussian spectrum (for GVD value of ϕ (2) = 0.1ps 2), (b) and as a function of both FWHM bandwidth and GVD.

Fig. 3
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 LP01 and LP02, respectively.

Fig. 4
Fig. 4

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

Fig. 5
Fig. 5

(a) and (b), reconstructed LP 01 and LP 02-mode (gamma-adjusted) at a center wavelength of 780 nm, (c) multi-path interference (MPI) values as a function of center wavelength of the bandpass.

Fig. 6
Fig. 6

(a) Spectrum without and after filtering with the 5-nm bandpass, (b) corresponding envelopes of the cross-correlation traces.

Fig. 7
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.

Fig. 8
Fig. 8

(a–c) Output of the fiber under test (near-field images) for different excitations, and corresponding changes in the cross-correlation trace.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

I ( x , y ) = Δ T / 2 + Δ T / 2 d t | E ( x , y , t ) | 2 = + d ω 2 π | E ( x , y , ω ) | 2 ,
E ( x , y , ω ) = m α m e m ( x , y , ω ) A m ( ω ) e i ϕ m + e r ( x , y , ω ) A r ( ω ) e i ϕ r .
ϕ m = β m ( ω ) L .
ϕ r = β r ( ω ) L r + ω c d ,
I ( x , y ) = I 0 ( x , y ) + I int ( x , y ) ,
I 0 = + d ω 2 π ( | e r A r ( ω ) | 2 + m | α m e m A m ( ω ) | 2 + ( m m ) 2 Re [ α m e m * A m * ( ω ) α m e m A m ( ω ) e i ( ϕ m ϕ m ) ] )
I int = m + d ω 2 π 2 Re [ e r * ( x , y , ω ) A r * ( ω ) α m e m ( x , y , ω ) A m ( ω ) e i ( ϕ m ϕ r ) ] .
Δ ϕ mr = Θ mr ( τ τ mr ) Ω + Δ ϕ mr ( Ω ) ,
I int ( x , y ) = m I m ( x , y , τ ) = m 2 α m Re [ e r * ( x , y , ω 0 ) e m ( x , y , ω 0 ) c mr ( τ τ mr ) exp ( i Θ mr ) ] ,
c mr ( t ) = 1 2 π d Ω S ( Ω ) exp ( i Δ ϕ mr ( Ω ) ) exp ( i Ω t ) .
I ( x , y ) = I 0 ( x , y ) + m 2 α m i r ( x , y ) i m ( x , y ) | c mr ( τ τ mr ) | cos ( ψ ) .
I m ( x , y , τ ) = α m | e m ( x , y ) e r ( x , y ) | S 0 Δ Ω π 1 ( 1 + d m 2 ) 1 / 4 exp [ ( τ τ m r ) 2 Δ Ω 2 4 ( 1 + d m 2 ) ] cos ( ψ ) ,
ψ = ϕ m ( x , y ) + Θ m r + ( τ τ m r ) 2 Δ Ω 2 4 ( 1 + d m 2 ) d m + const ,

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