1. Introduction

Coherent light scattering in space and time through disordered medium is a very interesting topic. Due to the inherent inhomogeneity within the amorphous medium through which light travels, the coherent scattering causes spatial and temporal speckle patterns. These speckles limit the resolution in various optical systems, such as the spatial resolution in optical imaging or data transmission rate in optical communication.

In terms of temporal scattering, there exists a set of modes called principal modes, or Eisenbud-Wigner-Smith eigenstates [1–3], that despite spatiotemporal scattering, always arrive at the output temporally unscattered. The existence of these principal modes in multi-mode fiber (MMF) was first proposed in 2005 [4], in analogy to the principal states of polarization (PSPs) in single-mode fiber [5, 6]. The principal modes represent the paths through the medium with the maximum possible spatial coherence [7]. A large amount of work ensued [8–11], but primarily focused on the theoretical investigation of the principal modes, in terms of modeling, numerical simulation and statistics on the mode delay distribution.

Experimentally creating and characterizing these special modes will have a major impact on understanding the coherent light scattering phenomenon, as well as higher order dispersion and nonlinearity of these principal modes [12,13]. It will also be the first step for any potential applications using the principal modes [14–17]. The only experimental demonstration of creating and characterizing principal modes in MMFs so far were only made recently [7, 18] due to the high accuracy required of the spatial and spectral information characterizing these highly scattering states. In these paper, the authors measured the full transfer matrix between the input and output of the MMFs, as well as its first order wavelength derivative, and then calculated the principal modes based on this information. On the other hand, since principal modes correspond to the temporal unscattered modes, it is possible to measure principal modes using exclusively temporal measurements. A mode dependent signal delay method was recently proposed [19]. There the authors showed that just by measuring the mean signal time delays for N² different predetermined inputs, it is possible to completely identify all the principal modes.

In this paper, we utilize the similarity between principal mode characterization and quantum state tomography [20,21], and derive a more straightforward algorithm for measuring principal states using the temporal delay of N² predetermined input modes. An analytic expression for the full mode delay operator based on these delays is given for an arbitrary N. We also show that if higher order dispersion is negligible, and the output is composed of discrete pulses, one only needs 3N − 2 inputs to fully identify all the principal modes. We will use numerical simulation to demonstrate how the methods work in a 3-mode fiber.

2. Group delay operator

The Maxwell’s equations, in the absence of sources and currents and under the paraxial approximation, are reduced to the Helmholtz equation [22]

For a narrow band optical signal centered at frequency ω, propagating in a multi-mode fiber (MMF) that supports N propagating modes, the light propagation due to mode coupling within the fiber can be described as [4]

In analogy to the principal states of polarization (PSPs) in single mode fibers (SMFs) [5], the principal modes in MMFs are defined such that given an input principal mode |pⁱⁿ〉, the corresponding output principal mode |p^out〉 is constant with respect to frequency ω to the first order, up to a certain phase, i.e.,

3. Tomography using mean delays

The first method we propose is based on the similarity between quantum state tomography using projection measurement [20,21] and operator tomography using mean value measurement. In quantum state tomography, the density matrix (a Hermitian matrix) is measured by projecting it onto a fixed set of selected input states. Here in the mean delay characterization, we measure the mode delay operator (a Hermitian matrix) by using the mean signal delays (equivalent to projection) of a set of input modes.

Below we detail the mathematics of the operator tomography method. An arbitrary Hermitian matrix in N dimension is in an N² dimensional vector space over the real number ℛ. Suppose we have N² Hermitian matrices {Γ_μ} (each one an N × N matrix) that satisfies

For the measurement of a physical quantity F̂ on a particular state |ψ_i〉, the mean value is

Here {M_i} matrices directly connect the measurement results f_i to the target matrix F. If we can efficiently calculate these {M_i} matrices, tomography of F is as simple as multiplying them by the measurement results {f_i} and add them up together, as indicated in Eq. (11).

Since each measurement provides one real number, and the total degrees of freedom for an arbitrary Hermitian matrix is N², one needs a minimum of N² different measurements.

Theorem1: B_iμ = Tr[Γ_μ · P_i] is reversible iff {P_i} are linearly independent

Proof: B_iμ = Tr[Γ_μ · P_i] is reversible means if ${\sum }_i{\alpha }_i{B}_{i\mu }=0$ for all μ, then α_i = 0.

The left hand side: for all μ, $0={\sum }_i{\alpha }_i{B}_{i\mu }={\sum }_i{\alpha }_i\text{Tr}\left[{\mathbf{\Gamma }}_{\mu }\cdot {\mathit{P}}_i\right]=\text{Tr}\left[{\mathbf{\Gamma }}_{\mu }\cdot {\sum }_i{\alpha }_i{\mathit{P}}_i\right]=\text{Tr}\left[{\mathbf{\Gamma }}_{\mu }\cdot \mathit{\alpha }\right]$

Then using Eq. (8), we have $\mathit{\alpha }={\sum }_{\mu }\text{Tr}\left[\alpha \cdot {\mathbf{\Gamma }}_{\mu }\right]{\mathbf{\Gamma }}_{\mu }={\sum }_{\mu }0\cdot {\mathbf{\Gamma }}_{\mu }=\mathbf{0}$.

Thus B_iμ is reversible is equivalent to if $\mathit{\alpha }\equiv {\sum }_i{\alpha }_i{\mathit{P}}_i=0$ then α_i = 0, which means {P_i} are linearly independent ▪

As a result, as long as we can choose N² input states {|ψ_i〉} so that the corresponding projection operators {P_i} are linearly independent, we can use Eq. (10) to calculate B_iμ and Eq. (12) to calculate {M_i} and Eq. (11) to calculate the matrix F. Here we propose a simple universal set of {|ψ_i〉} that has N² states and works for an arbitrary $N:\left\{|{\psi }_1^{(k)}〉,|{\psi }_2^{(k_1,k_2)}〉,|{\psi }_3^{(k_1,k_2)}〉\right\}$, where

Theorem2: {M_i} does not depend on the choice of {Γ_μ}, but only on the choice of the measurement set {P_i}

Proof: Suppose there is another complete set {Γ′_μ}. Since both {Γ_μ} and {Γ′_μ} are complete, they are connected by a linear reversible transformation

As a result, any {Γ_μ} that satisfies Eqs. (7a) and (7b) will work, and will generate the same {M_i}. Once again, we propose one universal set that works for an arbitrary $N:\left\{{\mathbf{\Gamma }}_{1,ij}^{(k)},{\mathbf{\Gamma }}_{2,ij}^{(k_1,k_2)},{\mathbf{\Gamma }}_{3,ij}^{(k_1,k_2)}\right\}$, where

As a matter of fact, since we have such simple analytic expressions for {|ψ_i〉} and {Γ_μ}, it turns out {M_i} also have simple analytic expressions for an arbitrary $N:\left\{M_{1,ij}^{(k)},M_{2,ij}^{(k_1,k_2)},M_{3,ij}^{(k_1,k_2)}\right\}$, where

To summarize, the process of using the mean signal delay method to find all the principal modes is as follows: first, one prepares N² different input modes {|ψ_i〉} according to Eq. (13). One can work under any orthonormal basis. One such basis could be the eigen modes of the fiber. Second, one launches these N² input modes into the fiber to measure their respective mean signal delay f_i. Third, one uses Eq. (19) to calculate {M_i} and Eq. (11) to calculate F. Last, one calculates the eigenvectors and eigenvalues of F, which are the principal modes under the chosen basis and their respective delays. To verify that these are indeed the principal modes, one can recreate the superposition states according to the eigenvectors, under the same basis, and launch them into the same fiber. These modes should experience no temporal spreading to the first order of mode dispersion.

We numerically simulate the tomography process in Appendix A for N = 3.

Error estimation

Here we consider the propagation of error during the input state preparation and mean signal delay measurement to the principal states calculation. In the ideal case with no error, the matrix for the group delay operator F̂(ω) (Eq. (11)) is reproduced here as

4. Tomography using projection operators

In section 3, we implement the full tomography on the operator using the mean signal delays. However, when the differences of group delays between principal modes are big enough, i.e., the output principal modes are temporally separated, we can resolve each principal mode component. In this case, the signal strength corresponding to the time delay of the j^th principal mode for the i^th input is

Assuming that the arbitrary principal state is decomposed in the eigen modes as

A numerical simulation of this method is shown in Appendix B.

5. Discussion

To experimentally implement our method, one needs a single spatial light modulator to launch the specified input states, and a fast photodetector to temporally resolve the optical signal. In the transfer matrix method, such as the one employed in [7] and [18], the phase information between different modes is retrieved through a reference beam in terms of homodyne detection. The interferometric stability degrades over long distance, and active stabilization might be necessary, making the setup more complicated. In our method, the phase information is encoded within the N² input states (self-referenced), thus no interferometer is needed.

The speed of the photodetector and the temporal width of the input pulse limit the temporal resolution, and thus the minimum measurable delay difference. If the mode delay differences of a group of principal modes are smaller than the temporal resolution, they can be regarded as degenerate, and any orthonormal basis in this subspace can be used as principal modes. Transmitting any one of these will result in minimal temporal delay spreading. Ultimately, the minimum pulse width (the inverse of bandwidth) is limited by higher order dispersion.

One thing we would like to point out here is that, the mode delay operator, though highly related, does not provide the full information of the transfer matrix, and vice versa. For example, one cannot calculate the transfer matrix from the mode delay matrix only. As a result, our method does not measure the output principal modes. If one need to identify both input and output modes, one needs to make additional measurements, by launching the input principal modes, and characterizing the corresponding output modes.

Compared with previous signal delay method [19], our method is more straightforward to derive. Since we now have an analytic expression for the mode delay operator, we no longer need to do the costly matrix inversion of a N² × N² matrix. Moreover, it prevents spreading numeric errors due to ill-conditioned matrices. Also, our definition of the mode delay operator F̂ has non-zero trace, where the trace is the mean signal delay. Therefore, we can measure the mean delay without additional measurements.

If the pulses are well seperated, our second method further reduces the number of necessary measurements from quadratic N to linear N. In cases where the measurement speed is critical, for example when the environment is changing quickly, which casues the principal modes to change quickly, this becomes a crucial saving.

Our methods are described in the context of mode coupling in multi-mode fiber. However, they are applicable to many cases, as long as all the spatial modes that are coupled together are included.

6. Conclusion

In this paper, we use the mathematical similarity between characterizing mode delay operator and quantum state tomography, and develop two new methods to characterize the principal modes. Our methods require smaller number of input states compared with mainstream methods, are more robust in terms of measurement errors, and should be more economical to implement.

Appendix A: Tomography using mean delays for N=3

Here, we numerically simulate the tomography process for N = 3 modes to illustrate our method.

We construct an arbitrary 3 × 3 Hermitian matrix for the group delay operator

(31)$$\mathit{F}=\left(\begin{array}{ccc}-1& 0.2i& 0.2\\ -0.2i& 0& 0.3+0.4i\\ 0.2& 0.3-0.4i& 0.5\end{array}\right)$$

We can calculate its eigenvalues, i.e., the mode delays of the principal modes {τ_k } (listed in Table 1). Our goal is to recreate the matrix F using measurement results of mean signal delay of chosen input states.

Table 1. Simulated temporal delays of the principal modes using mean delays

View Table | View all tables in this article

First, we construct N² = 9 launch modes according to Eq. (13) and get

$$\left\{|{\psi }_i〉\right\}=\left\{\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right),\left(\begin{array}{c}\frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}\\ 0\end{array}\right),\left(\begin{array}{c}\frac{1}{\sqrt{2}}\\ 0\\ \frac{1}{\sqrt{2}}\end{array}\right),\left(\begin{array}{c}0\\ \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right),\left(\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{i}{\sqrt{2}}\\ 0\end{array}\right),\left(\begin{array}{c}\frac{1}{\sqrt{2}}\\ 0\\ \frac{i}{\sqrt{2}}\end{array}\right),\left(\begin{array}{c}0\\ \frac{1}{\sqrt{2}}\\ \frac{i}{\sqrt{2}}\end{array}\right)\right\}$$

Second, we simulate the output signal of the i^th input signal according to

(32)$${\text{sig}}_i(t)=\sum _k{\left|〈p_k|{\psi }_i〉\right|}^2{r}_k(t)$$

where r_k (t) is the temporal response of k^th principal mode and assumed to be

(33)$$r_k(t)=\frac{1}{\sqrt{2\pi }{\sigma }_k}\text{exp}\left[-\frac{{\left(t-{\tau }_k\right)}^2}{2{\sigma }_k^2}\right]+∊$$

σ_k is the pulse width and is assumed to be 0.25. Each data point is also added a small error ∊ of Gaussian distribution with zero mean and standard deviation of 0.03. The simulated output signal is shown in Figure 1. The mean signal delay is then calculated as

(34)$$f_i^{\prime }=\frac{\int \text{d}t\cdot t\cdot {\text{sig}}_i(t)}{\int \text{d}t\cdot {\text{sig}}_i(t)}$$

and we get

$$\left\{f_i^{\prime }\right\}=\left\{\begin{array}{ccccccccc}-1.037& -0.014,& 0.539& -0.524,& -0.053,& 0.572,& -0.718,& -0.272,& -0.122\end{array}\right\}$$

Third, we calculate {M_i} according to Eq. (19) and get

$$\left\{{\mathit{M}}_i\right\}=\left\{\begin{array}{c}\left(\begin{array}{ccc}1& \frac{-1+i}{2}& \frac{-1+i}{2}\\ \frac{-1-i}{2}& 0& 0\\ \frac{-1-i}{2}& 0& 0\end{array}\right),\left(\begin{array}{ccc}0& \frac{-1+i}{2}& 0\\ \frac{-1+i}{2}& 1& \frac{-1+i}{2}\\ 0& \frac{-1-i}{2}& 0\end{array}\right),\left(\begin{array}{ccc}0& 0& \frac{-1+i}{2}\\ 0& 0& \frac{-1+i}{2}\\ \frac{-1+i}{2}& \frac{-1-i}{2}& 1\end{array}\right),\\ \left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right),\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 1& 0& 0\end{array}\right),\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right),\\ \left(\begin{array}{ccc}0& -i& 0\\ i& 0& 0\\ 0& 0& 0\end{array}\right),\left(\begin{array}{ccc}0& 0& -i\\ 0& 0& 0\\ i& 0& 0\end{array}\right),\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& -i\\ 0& i& 0\end{array}\right)\end{array}\right\}$$

Plug this into Eq. (11), the measured matrix

$${\mathit{F}}^{\prime }=\left(\begin{array}{ccc}-1.037& 0.002+0.192i& 0.196+0.023i\\ 0.002-0.192i& -0.014& 0.310+0.384i\\ 0.196-0.023i& 0.310-0.384i& 0.539\end{array}\right)$$

Compare this with our starting matrix F (Eq. (31)), the method reproduces the original matrix well.

 

Fig. 1 Simulated signal delay for 9 different input modes: ${\text{sig}}_i(t)={\sum }_k{\left|〈p_k|{\psi }_i〉\right|}^2{r}_k(t)$, where $r_k(t)=\frac{1}{\sqrt{2\pi }{\sigma }_k}\text{exp}\left[-\frac{{\left(t-{\tau }_k\right)}^2}{2{\sigma }_k^2}\right]+∊$. τ_k is its mode delay of k^th principal mode, σ_k = 0.25 is the pulse width, ∊ is the measurement error.

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Last, we calculate the eigenvalues {τ′_k} (listed in Table 1) and eigenvectors {|p′_k〉} of F′. The measurement errors on mode delays are approximately 2%. We also include the actual mean delays {τ″_k = 〈p′_k|F̂|p′_k〉} if these measured “principal states” are launched into the fiber, as well as the standard deviation of the mode delay σ′_k = (〈p′_k|F̂²|p′_k〉 − |〈p′_k|F̂|p′_k〉|²)^1/2. They are in the range of 0.03 – 0.04. In comparison, an arbitrary state has the standard deviation on the order of 1.

Appendix B: Tomography using projection operators for N=3

Here we numerically simulate the projection method. Since N = 3, our method should use 7 different input configurations for the complete tomography.

We assume the same mode delay matrix F as in Eq. (31). Also, since our selected input sets according to Eq. (23) is a subset of the inputs in the mean signal delay method, we use the same input data as well.

We global fit the simulated signal using ${\text{sig}}_i(t)={\sum }_{k=1}^3\frac{A_{i,k}}{\sqrt{2\pi }{\sigma }_k}\text{exp}\left[-\frac{{\left(t-{\tau }_k\right)}^2}{2{\sigma }_k^2}\right]$ to obtain mode group delay for the three principal modes {τ′_k} (listed in Table 3). The corresponding weights of each input A_i,k are listed in Table 2. During the fitting procedure, the parameters τ_k and σ_k are shared between different i. We also restrict all A_i,k to be non-negative, and ${\sum }_{k=1}^3{A}_{i,k}=1$ for i = 1 . . . 7.

Table 2. Extracted A_i,j from the global fitting

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Table 3. Simulated temporal delays of the principal modes using projection operators

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With these, we reconstruct the three principal modes using Eqs. (26) and (29), where we obtain

$$\left\{|p_k^{\prime }〉\right\}=\left\{\begin{array}{ccc}\left(\begin{array}{c}0.946\\ 0.027+0.262i\\ -0.184-0.041i\end{array}\right),& \left(\begin{array}{c}0.312\\ -0.165-0.801i\\ 0.426+0.230i\end{array}\right),& \left(\begin{array}{c}0.076\\ 0.505+0.120i\\ 0.665-0.532i\end{array}\right)\end{array}\right\}$$

Due to the measurement and numerical errors, the obtained principal modes are not exactly orthogonal to each other. For example, |〈p′₁|p′₃〉| = 0.017. To obtain the orthogonal principal modes, we recalculate F according to Eq. (30) and get

$${\mathit{F}}^{\prime }=\left(\begin{array}{ccc}-1.037& 0.002+0.192i& 0.196+0.023i\\ 0.002-0.192i& -0.014& 0.310+0.384i\\ 0.196-0.023i& 0.310-0.384i& 0.539\end{array}\right)$$

We also recalculate its eigenvalues {τ″_k} and eigenvectors {|p″_k〉} and again results are listed in Table 3.

Acknowledgments

The authors thank William A. Wood, Ioannis Roudas, and Robert Modavis for helpful discussions.

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