3. Effects of fiber geometry on spectral correlation of speckle

In this section, we present a theoretical analysis of the dependence of the spectrometer resolution on the fiber geometry. The fiber length L, the core diameter W, and the numerical aperture NA are crucial parameters determining the resolution and bandwidth of the fiberbased spectrometer. If we consider a monochromatic input light which excites all the guided modes, then the electric field at the end of a waveguide of length L is:

where A_m and φ_m are the amplitude and initial phase of the mth guided mode which has the spatial profile ψ_m and propagation constant β_m. To simplify the analysis, we considered planar waveguides of width W and numerical aperture NA, and calculated the mode profile and propagation constant using the method outlined in Ref [10]. We assumed that initially all of the modes in the waveguide were excited equally (Am = 1 for all modes) with uncorrelated phases (φm are random numbers between 0 and 2π). We calculated the spatial distribution of electric field intensity at the end of a waveguide, as a function of the input wavelength, and then computed the spectral correlation function of intensity C(Δλ). By repeating this process as we varied L, W, or NA, we were able to obtain the dependence of the spectral correlation width δλ on these parameters. Figure 2(a) plots δλ vs. L for W = 1 mm and NA = 0.22. δλ scales inversely with L, indicating that the spectral resolution can be changed simply by varying the waveguide length. Next we fix L = 1 m, and plot δλ as a function of W in Fig. 2(b) for NA = 0.22. The spectral correlation width initially drops, but quickly saturates. In Fig. 2(c), we varied NA while keeping L = 1 m and W = 1 mm, and observed a reduction of δλ with NA. To obtain a physical understanding of the simulation results, we present a qualitative analysis of speckle decorrelation using Eq. (1). Individual modes propagate down the waveguide with different propagation constants (βm) and accumulate different phase decays βm L). The maximum difference occurs between the fundamental mode (m = 1) and the highest-order mode (m = M), which we denote φ(λ) = β1(λ)L−βM(λ)L. In order for an input wavelength shift δλ to produce a distinct intensity distribution at the output, φ(λ) should change by approximately π, namely, |dφ(λ)/dλ| δλ ~π. By approximating β1(λ) ≈k and βM(λ) ≈k cos(NA) for large W (k = 2π n/λ, n is the refractive index of the waveguide, λ is the vacuum wavelength), we get δλ ~(λ/n)2/(2 n L)/[1 - cos(NA)]. For small NA, 1 - cos(NA) ≈(NA)2/2, and δλ ~λ2 / [n L (NA)2]. This simple expression captures the basic trends in Fig. 2(a-c). In Fig. 2(a), the spectral correlation width scales linearly with 1/L. When W is large, the spectral decorrelation does not depend on W [Fig. 2(b)]. Figure 2(c) is a log-log plot showing the calculated δλ fit well with a straight line of slope −2.09, thus confirming the 1/(NA)2 scaling. The above analysis assumed that the spectral correlation width was determined by the total range of the propagation constants in the waveguide, β1 - βM. This assumption led to the strong dependence of δλ on NA, which dictates the highest order mode the waveguide supports. The waveguide width W, which predominantly determines the β spacing of adjacent modes, had little effect on δλ. To validate this assumption, we considered a waveguide with fixed geometry (L = 1 m, W = 1 mm, and NA = 0.22) and calculated the spectral correlation function when exciting a subset of the ~300 guided modes (i.e. Am = 0 for some m’s). Figure 2(d) plots C(Δλ) in four cases: (i) all the modes are excited; (ii) only the 100 lowest order modes are excited; (iii) only the 100 highest order modes are excited; and (iv) every 5th mode is excited. The spectral correlation width decreased in (ii) and (iii), but remained virtually unchanged in (iv). These results confirm that the speckle decorrelation depends primarily on the difference between the maximal and minimal β values. Increasing W adds modes in between β1 and βM, but does not significantly modify the values of β1 and βM, thus having little impact on the spectral resolution. However, as will be discussed later, increasing W does increase the spectral bandwidth of the fiber spectrometer.

Fig. 2. (a) Calculated spectral correlation width δλ as a function of the length L of a planar waveguide with a fixed width W = 1 mm and NA = 0.22. The wavelength of the input light is 1500 nm. δλ scales linearly with 1/L (blue line), indicating that a longer waveguide provides finer spectral resolution. (b) Calculated spectral correlation width δλ as a function of the width W of a planar waveguide with a fixed length L = 1 m and NA = 0.22. The wavelength of the input light is 1500 nm. δλ decreases quickly at small W and then saturates at large W. (c) Calculated spectral correlation width δλ as a function of the numerical aperture NA of a planar waveguide with L = 1 m and W = 1 mm. The wavelength of the input light is 1500 nm. Red dotted line is a linear fit in the log-log plot of the calculated δλ (blue dots) vs. NA, it has a slope of −2.09, indicating δλ scales as 1/NA2. (d) Calculated spectral correlation function of the intensity at the end of a planar waveguide (L = 1 m, W = 1 mm, NA = 0.22), when a subset of the waveguide modes were excited. The waveguide supports ~300 modes at λ = 1500 nm. (i) All the modes were excited (black solid line), (ii) only the 100 lowest order modes were excited (blue dash-dotted line), (iii) only the 100 highest order modes were excited (green dashed line), (iv) every fifth mode was excited (red dotted line). The spectral correlation width decreased in the cases of (ii) and (iii), but remained virtually unchanged in (iv), indicating the speckle decorrelation is determined primarily by the total range of β values of the modes that are excited by the input signal.

Finally, we discuss the experimental implication of the above analysis. Figure 2(d) reveals that for a given fiber the fastest speckle decorrelation is reached only when we excite the guided modes that cover the full range of the propagation constants. Experimentally, the measured spectral correlation width of a given fiber changed by as much as ~50% depending on the coupling of the input, which could be varied by adjusting the lateral alignment of the single-mode fiber to the multimode fiber. In the future, chaotic multimode fibers might be used to avoid the alignment issue and ensure all of the modes are excited more or less equally [11]. The non-circular cross-section of the core, e.g. the D-shaped cross-section, leads to chaotic dynamics of light rays in the fiber. Consequently the majority of the guided modes are spread uniformly over the entire core, and they will be excited no matter where light enters the fiber at the input facet.