Super- and Anti-Principal Modes in Multi-Mode Waveguides

TL;DR

This paper introduces super- and anti-principal modes in multimode waveguides, which exhibit enhanced or reduced spectral correlations, outperforming traditional principal modes in stability and sensitivity, with potential applications in communication and imaging.

Contribution

The study presents a new class of states in multimode fibers that outperform principal modes in spectral stability and sensitivity, including the experimental generation and analysis of these states.

Findings

Super-principal modes have enhanced spectral stability.

Anti-principal modes are highly sensitive to frequency changes.

Modes are orthogonal even with mode-dependent loss.

Abstract

We introduce a new type of states for light in multimode waveguides featuring strongly enhanced or reduced spectral correlations. Based on the experimentally measured multi-spectral transmission matrix of a multimode fiber, we generate a set of states that outperform the established "principal modes" in terms of the spectral stability of their output spatial field profiles. Inverting this concept also allows us to create states with a minimal spectral correlation width, whose output profiles are considerably more sensitive to a frequency change than typical input wavefronts. The resulting "super-" and "anti-principal" modes are made orthogonal to each other even in the presence of mode-dependent loss. By decomposing them in the principal mode basis, we show that the super-principal modes are formed via interference of principal modes with closeby delay times, whereas the anti-principal modes are a superposition of principal modes with the most different delay times available in the fiber. Such novel states are expected to have broad applications in fiber communication, imaging, and spectroscopy.

Equations

$$q\hat{\varphi }=-i{t}^{-1}\left({\omega }_0\right)\frac{dt}{d\omega }{|}_{\omega ={\omega }_0}\hat{\varphi }=\tau \hat{\varphi },$$

$$\rho \left(\omega ,{\omega }_0\right):=-i{t}^{-1}\left({\omega }_0\right)\frac{t\left(\omega \right)-t\left({\omega }_0\right)}{\omega -{\omega }_0}.$$

$$\mathcal{T}\left(\hat{\varphi }\right)$$

Full text

I I. Introduction

Optical fibers are the enabling tools for transferring the ever-increasing data volumes characteristic for our information age. Since single-mode fibers will soon reach their capacity limit Richardson et al. (2013); Essiambre et al. (2010), the next step will be to switch to multi-mode fibers, whose spatial or transverse degree of freedom can be exploited for communication Kahn and Miller (2017); Randel et al. (2011); Ryf et al. (2012, 2011, 2013); Fontaine et al. (2015); Salsi et al. (2011); Carpenter et al. (2012); Carpenter and Wilkinson (2012) and imaging applications Flusberg et al. (2005); Choi et al. (2012); Plöschner et al. (2015); Čižmár and Dholakia (2012); Plöschner and Čižmár (2015); Čižmár and Dholakia (2011); Amitonova et al. (2016); Papadopoulos et al. (2013); Morales-Delgado et al. (2015a); Stasio et al. (2016); Loterie et al. (2017); Kolenderska et al. (2015). While many transverse modes in a fiber hold promise for a considerable increase in the information transmission capacity, the crucial challenge is to overcome the detrimental effects of random mode mixing, induced by fiber imperfections and external perturbations Raddatz et al. (1998); Patel and Ralph (2002); Shen et al. (2005); Panicker et al. (2008); Panicker and Kahn (2009); Morales-Delgado et al. (2015b); Parto et al. (2016); Carpenter et al. (2016a). In combination with the unavoidable modal dispersion, this leads to spatial and temporal distortion of optical signals, especially when the mode mixing is strong in the fiber.

A way out of this predicament is provided by the eigenstates of the Wigner-Smith time delay operator Wigner (1955); Smith (1960); Fan and Kahn (2005); Juarez et al. (2012); Shemirani et al. (2009); Ho and Kahn (2011); Rotter et al. (2011); Franz and Bulow (2012); Milione et al. (2015); Carpenter et al. (2015); Xiong et al. (2016, 2017); Carpenter et al. (2016b), also called the “principal modes” (PMs), which were introduced to suppress modal dispersion, allowing optical pulses to be sent through a multimode fiber (MMF) without distortion. In addition, the PMs maintain the spatial coherence so that the output field can be focused to a diffraction-limited spot or converted to any spatial pattern Carpenter et al. (2015); Xiong et al. (2016). The ability of maintaining temporal pulse shape and spatial coherence of a broadband signal amid strong spatio-temporal scattering is desirable for many applications including fiber communication, microscopy, MMF lasers and amplifiers Kim et al. (2012); Papadopoulos et al. (2012); Aulbach et al. (2011); Richardson et al. (2010); Jauregui et al. (2013); Florentin et al. (2017). Moreover, the PMs have great potential for quantum information protocols as they can be used to synthesize decoherence-free high-dimensional quantum states Peruzzo et al. (2011, 2010); Defienne et al. (2016). Thanks to the recent advances in wavefront shaping techniques Vellekoop and Mosk (2007); Vellekoop et al. (2010); Popoff et al. (2010a, 2011, b); Katz et al. (2011); Mosk et al. (2012); Rotter and Gigan (2017), the above desirable features of PMs are now also accessible in experiments using MMFs with weak or strong mode coupling Carpenter et al. (2015); Xiong et al. (2016, 2017).

In spite of their promising potential, PMs suffer from two major limitations. First, they have a finite spectral width, limiting the bandwidth of input signals that can maintain the temporal pulse shape and spatial coherence after transmitting through the MMF. This limitation is severe for MMFs with strong mode mixing, where the bandwidth of PMs is particularly narrow. Secondly, PMs are non-orthogonal to each other in a MMF with mode-dependent loss (MDL), resulting in crosstalk between them. A fundamental question is thus whether an orthogonal set of states exist that outperform PMs in terms of the spectral correlation width, especially in the strong mode coupling regime where a gain in bandwidth is in the highest demand. If such special states indeed exist, a practical question is how to generate them experimentally - an operational procedure which could have a broad impact on fiber applications in communication, imaging, nonlinear microscopy, laser amplifiers, and quantum technology.

In a seemingly unrelated context, it has recently been shown that an individual multimode fiber can function as a spectrometer with ultrahigh resolution, broad bandwidth and low loss Redding and Cao (2012); Redding et al. (2013, 2014a, 2014b, 2016); Liew et al. (2016); Coluccelli et al. (2016); Valley et al. (2016). The output speckle pattern of this MMF is formed by interference of the guided fiber modes, and it changes with input frequency. The working principle of the fiber spectrometer is now to use the frequency-dependent speckle pattern as a fingerprint to recover the input spectrum. To further increase the resolving power of this device, one needs to enhance the frequency sensitivity of its output speckle patterns. The key question here is whether it is possible to achieve this by creating a special input state whose output field pattern is considerably more sensitive to a frequency change than that of a typical input to the MMF. The crucial point is, in other words, whether one can use the spatial degree of freedom at the input to accelerate the spectral decorrelation at the output of a MMF.

A practical application of MMFs in biomedical imaging is to reduce the spatial coherence of a broadband light source for parallel optical coherence tomography (OCT) Dhalla et al. (2010). When imaging through turbid media such as biological tissue, the suppression of spatial coherence of the illuminating light prevents resolution loss from crosstalk due to coherent multiple scattering. If it was possible to create a state in the MMF with much reduced spectral correlation, one could greatly suppress the spatial coherence at the MMF output by launching broadband light into such a state.

Here we introduce such novel states of light that have the aforementioned unique characteristics and we generate them experimentally by addressing the spatial degrees of freedom of a MMF with wavefront shaping techniques. As we demonstrate explicitly, exceeding the already efficient performance of PMs is possible by making use of the information stored in the multi-spectral transmission matrix of a MMF. Based on an optimization procedure operating on such a matrix Andreoli et al. (2015), we are able to create a set of “super-PMs” that not only have a significantly increased spectral stability as compared to the most stable PM available in the same fiber, but that also have the great advantage of being mutually orthogonal even in the presence of MDL. Moreover, our optimization procedure can be applied to generate “anti-PMs” featuring an extremely narrow spectral correlation width – a property that holds promise for spectroscopy and sensing applications with MMFs.

Intuitively, the physical mechanisms for the formation of both “super-PMs” and “anti-PMs” can be understood in the basis of PMs. In the absence of MDL, “super-PMs” are formed by PMs with similar delay times and the interference effect between them brings about a broader bandwidth for the “super-PMs”. On the contrary, “anti-PMs” are composed of PMs with extremely different delay times, which directly results in a bandwidth even narrower than that of random inputs. More importantly, our analysis illustrates that the PMs provide a powerful basis for synthesizing new types of states with unique spatial, temporal and spectral characteristic.

II II. Principal Modes

Principal modes $\hat{\phi}$ are the eigenstates of the Wigner-Smith time-delay operator $q$ for a MMF with transmission matrix $t(\omega)$,

[TABLE]

where $\tau$ is the respective complex eigenvalue whose real part is the so-called proper delay time Fan and Kahn (2005); Carpenter et al. (2015); Xiong et al. (2016); Ambichl et al. (2017). From the above construction it directly follows that PMs do not suffer from modal dispersion to the first order in frequency change (see supplementary material Ambichl et al. (2017)), such that the output field pattern does not vary with frequency in the vicinity of $\omega_{0}$. A suitable measure for the modal dispersion, i.e., the deviation of the output vector $\vec{\psi}\!\left(\omega\right)=t\!\left(\omega\right)\hat{\phi}$ from its respective direction at $\omega_{0}$, is the field autocorrelation function $C\!\left(\omega,\omega_{0}\right):=|\hat{\psi}^{\dagger}\!(\omega)\cdot\hat{\psi}(\omega_{0})|\in\left[0,1\right]$, where $\hat{\psi}\!\left(\omega\right)=\vec{\psi}\!\left(\omega\right)\!/|\vec{\psi}\!\left(\omega\right)|$ is the normalized output vector. We define the correlation width of a given input state as the spectral range surrounding $\omega_{0}$ where $C(\omega,\omega_{0})\geq 0.9$ Xiong et al. (2016).

Experimentally we tested a one-meter-long step-index MMF with a core diameter of 50 $\mu$m and a numerical aperture of 0.22. It features about 120 linearly polarized (LP) modes at $\lambda_{0}=1550$ nm for one polarization. To directly access the regime of strong mode coupling, we apply stress to the fiber through external clamps. A tunable laser is employed for measuring the transmission matrix at different frequencies. Figure 1(a) shows our off-axis holographic setup for measuring the light field transmitted through the MMF. We use a spatial light modulator (SLM), which is imaged to the input facet of the MMF, to scan the incident angle of monochromatic laser light. For each input wavefront, the output signal from the fiber combines with the reference arm, forming interference fringes on the camera. We extract the amplitude and phase of the field from the interference pattern.

After recording the multi-spectral transmission matrix $t\!\left(\omega\right)$, we truncated the matrix to reduce noise by discarding the eigenchannels with very high loss (90 independent transmission channels were retained out of the 120 available ones). Based on the truncated transmission matrix, we compute the time-delay matrix $q$ and find its eigenstates (PMs). Applying a computer-generated phase hologram to the SLM, we create the PMs by constructing their input states with amplitude and phase modulations Arrizón et al. (2007); Bagnoud and Zuegel (2004). The output field patterns are recorded as the input frequency is scanned. Figure 1(b) shows the measured auto-correlation function $C(\omega,\omega_{0})$ for the PM with the largest correlation width (blue solid line), which has about three times the width as compared to that of a random input wavefront (gray dashed line).

III III. Super-Principal Modes

The fundamental question we address here is whether it is possible to increase the correlation width beyond the values obtained for PMs. To achieve this goal, we revisit the design principle of PMs, which are generated by the operator $q$ in Eq. (1) such that their output field patterns do not change when changing the input frequency incrementally by $d\omega$ from the reference value $\omega_{0}$ (see supplementary material Ambichl et al. (2017)). In a first step we modify this approach to produce a state whose output pattern stays invariant when the input frequency is changed by a finite shift $\Delta\omega$ rather than by the infinitesimal value $d\omega$.

To implement such a strategy, we introduce a new operator that aligns the output vectors of its eigenstate at two arbitrarily spaced frequencies $\omega_{0}$ and $\omega$ (with $\omega-\omega_{0}=\Delta\omega$). This new operator $\rho\!\left(\omega,\omega_{0}\right)$ takes the following form,

[TABLE]

It approaches the time-delay operator $q$ for small frequency spacing between $\omega_{0}$ and $\omega$, i.e., $q=\lim_{\Delta\omega\rightarrow 0}\rho(\omega,\omega_{0})$. An eigenstate of this new operator $\rho\!\left(\omega,\omega_{0}\right)$ features an output correlation function $C\!\left(\omega,\omega_{0}\right)$, which peaks not only at $\omega_{0}$ but also at $\omega$. With a gradual increase of $\left|\Delta\omega\right|$, the two correlation peaks move further apart, suggesting the possibility of extending the correlation bandwidth beyond that of the PM. We thereby use the term “super-PM” to label a state that has a correlation bandwidth of the output field pattern exceeding that of the widest PM in a given fiber.

From the experimentally measured transmission matrix $t\!\left(\omega\right)$, we numerically generate the super-PMs as the eigenstates of $\rho\!\left(\omega,\omega_{0}\right)$ in Eq. (2), and find the anticipated two peaks at $\omega$ and $\omega_{0}$ in the correlation function, as shown by the green dotted curve in Fig. 1(b). The width of each peak is given approximately by the spectral correlation width $\delta\omega$ of the transmission matrix (i.e., the correlation width associated with random input). As the frequency spacing between the two peaks at $\omega$ and $\omega_{0}$ exceeds the peak width $\delta\omega$, a dip develops in between the two peaks. To maximize the correlation bandwidth, we choose the spacing $\omega-\omega_{0}$ such that $C(\omega,\omega_{0})$ drops to the threshold value of 0.9 in between the two peaks in Fig. 1(b). In this case, the correlation width is about $18\%$ wider than the width of the widest PM. A further increase of $\Delta\omega$ makes the dip in between the two peaks drop to below $0.9$, causing a sudden decrease of bandwidth.

To reach the ideal case of a flat correlation function $C\!\left(\omega,\omega_{0}\right)\approx 1$ over the entire interval $[\omega_{0},\omega]$, $\Delta\omega$ must be less than $\delta\omega$. The fundamental reason that prevents such a flat correlation curve from being realized in a broader frequency interval can be understood from the construction principle of $\rho\!\left(\omega,\omega_{0}\right)$ and of its eigenstates: a state with a flat correlation function $C\!\left(\omega,\omega_{0}\right)\equiv 1$ in a finite interval $[\omega_{0},\omega]$ would have to be a simultaneous eigenstate of all operators $\rho\!\left(\omega,\omega_{0}\right)$ in this $\omega$-interval. The relevant commutator that would have to vanish for this to be possible is, however, non-zero in general: $[\rho(\omega_{1},\omega_{0}),\rho(\omega_{2},\omega_{0})]\neq 0$ with $|\omega_{1}-\omega_{2}|>\delta\omega$.

In spite of this restriction, we will now show how to create a set of states with correlation bandwidth considerably exceeding that of conventional PMs. In addition, we will make such “super-PMs” mutually orthogonal, even when the fiber has MDL. Both the PMs and super-PMs described above are non-orthgonal, as the operators $q(\omega_{0})$ and $\rho\!\left(\omega,\omega_{0}\right)$ become non-Hermitian in the presence of MDL. We will thus work with a new approach that constructs the desired set of super-PMs as approximate rather than as perfect simultaneous eigenvectors of the operators $\rho\!\left(\omega,\omega_{0}\right)$ within a desired spectral interval. Achieving this objective calls for the implementation of an optimization procedure with a non-linear cost function involving the multi-spectral transmission matrix $t\!\left(\omega\right)$ and the operators $\rho\!\left(\omega,\omega_{0}\right)$. Given the dimensionality of the problem determined by the number of fiber modes and the width of the spectral region of interest, such an optimization is a very demanding task even numerically.

In the following we take a shortcut to create orthogonal super-PMs by working with a considerably reduced optimization functional $\mathcal{T}$ (cost function),

[TABLE]

The spectral range that is effectively taken into account in this functional is defined by the weighting function $W\!\left(\omega\right)$, which is chosen to be a step-like, but continuous function centered around $\omega_{0}$, see Fig. 2(a) (purple dash-dotted curve). To maximize the width of the correlation function $C\!\left(\omega,\omega_{0}\right)$, the (scalar) value of $\mathcal{T}$ needs to be minimized depending on the input state $\hat{\phi}=t^{-1}\!\left(\omega\right)\vec{\psi}\!\left(\omega\right)$. We implement an efficient minimization of $\mathcal{T}$ using a simple gradient-based scheme (see supplemental material Ambichl et al. (2017) for the exact expressions of $W(\omega)$ and $\delta\mathcal{T}/\delta\hat{\phi}$). We note that our optimization scheme is a purely numerical optimization based on the experimentally measured transmission matrices rather than an experimental feedback loop.

We typically obtain the widest super-PM with an optimization loop that is initiated with the widest PM as a starting point for the corresponding iteration. Next, a whole cascade of optimizations of Eq. (3) is carried out, where in each optimization loop for a single super-PM the cost function $\mathcal{T}$ is minimized in the vector space orthogonal to each of the super-PMs obtained in the preceding steps. In this way, a strictly orthogonal set of super-PMs is obtained.

The input states for these super-PMs are created by the SLM and injected into the fiber. Figure 2(a) shows the widest super-PM (green dotted line). Its spectral correlation width is about $70\%$ larger than that of the widest PM. Out of the total 90 modes in the fiber after the matrix truncation, we find 20 super-PMs that outperform the widest PM in terms of the correlation width. Figure 2(b) shows the spectral width of super-PMs normalized by that of random input wavefronts. The widest super-PM has a bandwidth increase of 4.5 times as compared to the random input. In addition, we also plot the 20 PMs with largest bandwidth and find that the bandwidth enhancement for super-PMs is well above that of the PMs.

Due to fiber absorption and scattering loss, the PMs are non-orthogonal already at the fiber input, since they are the eigenstates of the non-Hermitian operator $q$. The propagation through the fiber will further increase this non-orthogonality, thus PMs are non-orthogonal both at the fiber input and output ends. By contrast, the super-PMs, in the way we construct them, are orthogonal at the input. We also observe that the orthogonality of the super-PMs at the input facet is well sustained during propagation through the fiber. The overlap of the output field patterns is defined as $O_{mn}:=|\hat{\psi}^{\dagger}_{m}\left(\omega_{0}\right)\cdot\hat{\psi}_{n}\left(\omega_{0}\right)|$ for the $m$-th and $n$-th super-PM. For a perfectly orthonormal set of vectors we would have $O_{mn}=\delta_{mn}$, where $\delta_{mn}$ is the Kronecker delta. The 20 widest PMs we consider are rather far away from this ideal case, as shown in Fig. 2(c). The average off-diagonal element of $O$ is 0.188. The super-PMs preserve their initial orthogonality to a large extent as shown in Fig. 2(d). On average we find for the off-diagonal elements of $O$ a reduced value of $0.068$. Thus the crosstalk between different super-PMs is strongly suppressed as compared to the PMs.

To validate our initial hypothesis that the enhanced bandwidth of super-PMs is linked to the property that they mimic mutual eigenstates of all $\rho\!\left(\omega,\omega_{0}\right)$ operators in the spectral region of interest, we work out a corresponding measure that quantifies how close an input state is to a simultaneous eigenvector of these operators (see supplemental material Ambichl et al. (2017) for details). Applying this measure to the PMs and to the super-PMs, indeed, shows that the super-PMs are much closer to mutual eigenstates of $\rho\!\left(\omega,\omega_{0}\right)$ than the conventional PMs, despite the fact that the information on $\rho\!\left(\omega,\omega_{0}\right)$ is included in the cost function $\mathcal{T}$ only implicitly (see supplemental material Ambichl et al. (2017) for the corresponding numerical values we obtained).

Can we also understand on a more intuitive level why and how super-PMs manage to outperform PMs? To answer this question we first decompose both the PMs as well as the super-PMs realized in the experiment in the basis of LP modes of the fiber. In the regime of strong mode coupling where all of the results shown above were obtained, we find that both PMs and super-PMs consist of nearly all LP modes, and the higher-order LP modes have smaller contributions due to higher loss (see supplemental material Ambichl et al. (2017)). While this finding confirms that PMs as well as super-PMs are really non-trivial combinations of fiber modes in the strong-coupling regime Xiong et al. (2016), it does not shed any light on the difference between PMs and super-PMs. We therefore change basis from the LP-modes to the PMs themselves, which constitute the natural basis to capture the dynamical aspects of light scattering as each PM is associated with a proper delay time. Also their greatly reduced wavelength dependence makes PMs very suitable to describe the transmitted light in a broad frequency window. When decomposing the super-PMs in the basis of PMs (with the bi-orthogonal basis vectors being sorted according to the delay times of the PMs), we observe clearly that the super-PMs are composed of PMs with neighboring delay times, as shown in Fig. 3(a).

The physical interpretation of this result is that super-PMs build on a very narrow distribution of delay times and enhance the spectral correlation width via interference of several time-delay eigenstates. To demonstrate that the relative phases of constituent PMs are essential, we also plot in Fig. 3(b) the correlation functions after adding random phases to the decomposition coefficients of the PMs (while maintaining the absolute magnitudes). It is evident that randomizing the phase significantly decreases the correlation width. The sensitivity to the relative phase of constituent PMs indicates that the interference of the PMs involved in the formation of a super-PM is essential.

Another relevant aspect is that the mixing of neighboring PMs to create super-PMs requires that a sufficient number of PMs are available with similar delay times. In a MMF with weak mode coupling, the values of the delay times are spread over a much broader time range, and hence the super-PM is composed of only one or two PMs with neighboring delay times. Therefore, the bandwidth of a super-PM is typically not significantly different from its constituent PMs for weak mode coupling. To test this conjecture, we apply the same optimization algorithm as above to fibers in the weak mode coupling regime. The corresponding results indicate that in this case the super-PMs are very similar to individual PMs and no significant admixture from several PMs is seen. Consequently, the correlation width of even the widest super-PM barely exceeds that of the widest PM (see the supplemental material Ambichl et al. (2017)). Super-PMs thus realize their full potential in the regime of strong-mode coupling where a gain in bandwidth is also most relevant since the PM-bandwidths in the strong-coupling limit are much narrower than for weak mode-coupling Xiong et al. (2017).

IV IV. Anti-Principal Modes

We will now investigate whether novel states of light may also be found when turning the concept of principal modes on its head. Specifically, we will address the question, whether it is possible to create not only states with a very broad spectral correlation width such as the PMs or the super-PMs studied above, but also states with a drastically reduced bandwidth as compared to the values associated with a typical or random input wavefront. Such “anti-PMs” would have an enhanced frequency sensitivity as desired, e.g., for fiber-based spectrometers whose operation principle relies on a very narrow correlation bandwidth Redding et al. (2013); Redding and Cao (2012). The optimization algorithm presented above now allows us to generate such highly sensitive states by just maximizing instead of minimizing the functional $\mathcal{T}$ in Eq. (3).

We first apply the algorithm for anti-PMs on an ideal MMF with no MDL. The corresponding unitary transmission matrix is obtained numerically with the concatenated fiber model developed earlier Ho and Kahn (2011). For simplicity, we consider a planar waveguide with a core width of 200 $\mu m$ and a numerical aperture of 0.22, supporting 57 guided modes Okamoto . The total length of the waveguide is one meter, divided into 20 segments in each of which light propagates without mode coupling. Between adjacent segments, all modes are randomly coupled, as simulated by a unitary random matrix Xiong et al. (2017). The results obtained by applying the optimization algorithm to this numerical model are shown in Fig. 4(a), displaying a significant reduction of the spectral correlation width of the anti-PM as compared to random inputs. Moreover, when decomposing the anti-PM in the basis of PMs, we immediately see that the narrowest anti-PM consists mainly of admixtures between the fastest and slowest PMs [see Fig. 4(b)]. These PMs have the largest achievable temporal difference in the MMF and mixing them with similar weight thus leads to an efficient decorrelation when changing the input frequency. Quite intuitively, the anti-PMs thus not only form the antipodes of the super-PMs in terms of their correlation bandwidth, but also in terms of the way they are constructed: while super-PMs group together several PMs with similar time delay, the anti-PMs combine those PMs with the most different time delay available. Whereas the relative phase with which this superposition of PMs is implemented clearly matters for super-PMs [see Fig. 3(b)], we find that for anti-PMs the correlation bandwidth is barely affected when changing the phases of constituent PMs [see the red dashed curve in Fig. 4(a)].

The reason behind this observation is that for PMs associated with very different delay times adjusting their relative phase cannot mend these states’ intrinsic tendency to decorrelate with frequency, while for states with nearby delay times the appropriate phases will optimize the interferences for specific mode-coupling in the MMF such as to increase the correlation width. As shown by the red dashed curve in Fig. 4(a), the correlation function features two enhanced side peaks adjacent to the main peak for the anti-PM with phases of constituent PMs randomized. After reaching zero, the correlation revives as $\omega$ is tuned further from $\omega_{0}$. This revival is caused by the beating of the slowest and fastest PMs, the two main components of the anti-PM. When the relative phase of the fastest and slowest PMs accumulates 2$\pi$, the output field should be the same as the one when the relative phase is zero. However, small contributions from PMs with intermediate delay times give rise to multi-path interference that suppresses the side peaks in the anti-PM in a way that is phase-sensitive.

To connect with the experimental case of anti-PMs, it is essential to take into account the mode dependent loss (MDL) in the fiber. We gradually increase the MDL in the numerical simulations up to the value close to the MMF used in the experiment. The narrowest anti-PM displays a narrowing of the spectral correlation function, and its bi-modal composition in the PM-basis changes drastically (see supplemental material Ambichl et al. (2017)). In particular, the increase of MDL enhances the weight of the slowest PMs, and the composition of the anti-PM in the PM-basis becomes much broader. This can be explained by considering that the MDL attenuates the slowest PMs most strongly since they stay longest inside the fiber. The broadening of the PM-composition can then be understood as a mechanism to compensate for the reduced contributions from the slow PMs. Since the constituent PMs with similar delay times overlap in time, their relative phases become important. When randomizing their phases, the bandwidth of anti-PMs is increased (see supplemental material Ambichl et al. (2017)).

When generating the anti-PMs in the MMF we find 62 anti-PMs out of 90 fiber modes, whose spectral correlation width is narrower than that of random inputs. Fig. 5 shows the data for the narrowest anti-PM. Due to the MDL, it is broadly spread in the PM basis, but it still decorrelates much more quickly than the random input with frequency detuning. Only when moving from the strong to the weak mode-coupling limit do we find that anti-PMs have about the same correlation width as a random input even without MDL (see supplemental material Ambichl et al. (2017)). This is because in the weak mode coupling regime, the fastest PM and the slowest PM have very different delay times. The beating between the two PMs is so strong that it causes significant side peaks of the spectral correlation function. To reduce the side peaks, additional PMs with intermediate delay times must be included. The inclusion of many intermediate PMs makes the correlation width of the anti-PM similar to that of a random input. Therefore, both super-PMs and anti-PMs have in common that they unfold their full potential in the limit of strong mode coupling.

V V. Summary

In summary, we present a new set of light states in multi-mode fibers that have either a significantly broader spectral correlation than the principal modes or a significantly narrower spectral correlation than a random input wavefront, respectively. We thus term these new states super- and anti-principal modes, and demonstrate how to generate them with a simple gradient-based algorithm based on the experimentally measured multi-spectral transmission matrix. Our optimization algorithm allows us to determine a whole set of such super- or anti-PMs, which are mutually orthogonal to each other even in the presence of mode-dependent loss and thus feature much reduced crosstalk. By overcoming the limitations of PMs, super-PMs outperform the PMs in terms of bandwidth and orthogonality. The performance gain is the highest in the regime of strong mode coupling as for very long fibers, where an increase in bandwidth is also most sought-after.

Moreover, we provide a physical understanding of how these states are formed by decomposing them in the PM basis. Super-PMs tend to combine several PMs with nearby delay times in a super-position with optimized phases. Anti-PMs, on the contrary, tend to combine PMs with the most different delay times available in the fiber. The presence of MDLs leads to modifications of these states, which are analyzed in detail.

On the one hand, the large bandwidth and mutual orthogonality of super-PMs pave the way for their application to dispersion-free transmission of light pulses through multi-mode fibers. On the other hand, the high spectral sensitivity of anti-PMs is ideally suited for optimizing the resolution of fiber-based spectrometers as well as for reducing the spatial coherence of a broadband light source for crosstalk-free imaging. Furthermore, this work illustrates the potential of using PMs to synthesize novel light states beyond the ones demonstrated here.

VI Acknowledgment

We acknowledge Chia Wei Hsu, Joel Carpenter, and Nicolas Fontaine for helpful discussions. This work is supported partly by the U.S. National Science Foundation under the Grant No.ECCS-1509361 and by the US Office of Naval Research under the MURI grant No. N00014-13-1-0649. P. A. and S. R. acknowledge support by the Austrian Science Fund (FWF) through projects SFB NextLite (F49-P10) and by the European Commission under project NHQWAVE (GA number 691209).