Multi-path interference (MPI) is a fundamental concept for understanding interac- tions between coherent modes in fibers. As discussed in the previous section, each mode is a discrete state with a unique propagation constant β. As such, when multiple modes with different β values are excited in the fiber, this creates multiple transmis- sion paths with different relative delays (as shown in Fig. 2·8). Consequently, the fiber effectively becomes an in-line Mach-Zender interferometer. The electric field at the end of the fiber is given by
E(r, φ, ω) =X
l
X
m
αl,mψl,m(r, φ)ei(ωt−βl,mL) (2.22)
where l and m are the azimuthal and radial eigenvalues of some LP mode, αl,m is
the relative amplitude of the mode, ψl,m is the transverse electric field, βl,m is the
propagation constant, ω is frequency, t is time, and L is the length of the fiber. First, consider the case where only two modes, mode a and mode b, have significant power within the fiber. The electric field has the form
E(r, φ, ω) = E0ψa(r, φ)e−iβaL+ αE0ψb(r, φ)e−iβbL eiωt (2.23)
and the intensity distribution I(r, φ, ω) = |E(r, φ, ω)|2 is given by
Figure 2·8: Schematic illustrating MPI in a few-mode fiber. (a) An arbitrary intensity pattern is incident on the fiber, exciting LP0,1 along
with parasitic LP1,1 and LP0,2 content. Each mode has a different
relative delay in the fiber – thus there is interference at the output of the fiber. (b) Wavelength-dependent intensity at the output of the fiber. There are two characteristic frequencies evident from interference between the dominant LP0,1 mode and each of the parasitic LP1,1 and
LP0,2 modes.
where E0 is the amplitude of mode a, αE0 is the amplitude of mode b, Ψa and Ψb are
the intensities of modes a and b respectively, and ∆β is the difference in propagation constants βb− βa. The intensity at a point in space (r, φ) will oscillate as a function
of frequency given that β = ω/c · nef f(ω), where nef f is the effective index of the
mode. The modulation depth of the interference will depend on the amplitudes of the interfering fields such that equal intensity distribution will lead to the largest fringes. This interference pattern will be evident depending on the conditions of the measurement. For example, a camera image of the ensemble will show a coherent superposition (depending on the coherence length of the source, as discussed below) of the modes dependent on the relative phase between them, by virtue of the fact that the device’s pixels ensure spatial resolution is preserved. Alternatively, if the entire beam is collected in a manner that does not preserve spatial information (i.e. a power detector or “single pixel” measurement) then the resulting power is equivalent to an integration of Eq. (2.24) over all space. Given that all of the power is collected for
this case, frequency-dependent oscillations will not be observed. This is an interesting caveat in that MPI is not an issue for systems where the entirety of the mode content can be captured. However, if the detection system is “mode selective” in that not all of the power from all of the modes is collected, or the measurement preserves spatial resolution, then interference can be observed. Assuming the coupling system is mode selective, an optical spectrum analyzer (OSA), for example, will show interference fringes for broadband spectra. MPI can also manifest as noise when detecting power at the output of the fiber if one does not take care to collect all of the power exiting the waveguide.
To determine the explicit wavelength dependence of the interference pattern at some coordinate (r, φ), we take the derivative of ∆β with respect to wavelength
d∆β dλ = − d dλ 2π λ ∆nef f(λ) = 2π λ2 ∆nef f − λ d dλ(∆nef f) (2.25)
The quantity ∆nef f − λdλd∆nef f is defined as ∆ng, the group delay difference be-
tween the modes. Group delay is a common physical parameter for determining the wavelength-dependent delay of modes in optical fibers. From Eq. (2.25), it is straightforward to show that the resulting fringe spacing of the interference pattern as a function of wavelength is given by
∆λ = λ
2
∆ng(λ)L
(2.26)
The fringe spacing corresponds to a characteristic wavelength-dependent oscillation in the intensity of the light leaving the fiber at some point in space, which is di- rectly proportional to ∆ng between the modes. Resolving these interference fringes
is dependent on both the bandwidth of the source and wavelength-resolution of the measurement device.
evant parameter to consider is the coherence length of the light source given by Lc ∼ λ20/∆λB, where λ0 is the center wavelength, and ∆λB is the bandwidth. The
coherence length represents the longest path length difference for which the wave- length components in the source remain coherent with respect to one another. The path length difference between the modes in a fiber is given by ∆ngL where L is the
length of the fiber. If ∆ngL > Lc, coherence is degraded and interference can not
be observed on the camera. As an example, consider two modes with a group index difference of 0.5×10−4. If the imaging source is a light emitting diode (LED) (λ
0 =
1000 nm, ∆λB = 20 nm), then the modes can propagate through approximately 1 m
in the fiber before interference is no longer observable. If instead, the imaging source is an external cavity diode laser (ECL) (λ0 = 1000 nm, ∆λB = 0.33 fm), then the
modes can propagate up to ∼3 km and remain coherent.
If interference is measured with wavelength-resolved means, such as an OSA, then coherence is dictated by the resolution of the instrument. Provided that the resolution of the instrument is high enough to sample the fringe spacing above the Nyquist limit (δλ < ∆λ/2, where ∆λ is the fringe spacing given by Eq. (2.26)), then interference can be resolved (Nyquist, 1924).
It can be shown that, for a given frequency, the peak-to-peak fluctuations of the intensity on a dB scale (denoted ptp) are related to the relative intensity of the parasitic mode, α2, by (Ramachandran et al., 2003b)
α2 = 10 ptp 20 − 1 10ptp20 + 1 !2 (2.27)
The interference between two modes leads to a characteristic frequency with a given depth of fluctuation dependent on the relative powers between the modes. It follows that if more modes are excited in the fiber, more frequencies will be apparent in the intensity pattern. For example, Fig. 2·8(b) shows the interference between a dominant
LP0,1 mode and the LP1,1 and LP0,2 modes, which are assumed to have significantly
less power. There are two characteristic frequencies resulting from each low-power, parasitic mode interfering with the fundamental mode.2
On one hand, MPI can be useful for measuring the mode content in a fiber. In Chap. 4, an experimental technique is demonstrated for measuring wavelength- dependent interference in order to calculate the modal power distribution in a given fiber. Purity analysis techniques like these are indispensable for operating in the highly multi-mode regime. On the other hand, MPI can have serious drawbacks, particularly for fiber lasers (Wielandy, 2007). The output electric field of a multi-mode fiber is a coherent superposition, dependent on the phase between the constituent modes which can fluctuate due to changes in ambient temperature, fiber layout, or vibration, etc. Thus the pointing stability, peak intensity, and beam profile of the laser are subject to random variations and degradation.