94340

All-fiber spectrometer based on speckle pattern reconstruction

Научная статья

Физика

We investigate the effect of the fiber geometry on the spectral resolution and bandwidth and also discuss the additional limitation on the bandwidth imposed by speckle contrast reduction when measuring dense spectra. Introduction While traditional spectrometers are based on onetoone spectraltospatial mapping spectrometers can also operate on...

Английский

2015-09-08

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All-fiber spectrometer based on speckle pattern

reconstruction

Brandon Redding, Sebastien M. Popoff, and Hui Cao*

Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA

*hui.cao@yale.edu

Abstract: A standard multimode optical fiber can be used as a general purpose spectrometer after calibrating the wavelength dependent speckle patterns produced by interference between the guided modes of the fiber. A transmission matrix was used to store the calibration data and a robust algorithm was developed to reconstruct an arbitrary input spectrum in the presence of experimental noise. We demonstrate that a 20 meter long fiber can resolve two laser lines separated by only 8 pm. At the other extreme, we show that a 2 centimeter long fiber can measure a broadband continuous spectrum generated from a supercontinuum source. We investigate the effect of the fiber geometry on the spectral resolution and bandwidth, and also discuss the additional limitation on the bandwidth imposed by speckle contrast reduction when measuring dense spectra. Finally, we demonstrate a method to reduce the spectrum reconstruction error and increase the bandwidth by separately imaging the speckle patterns of orthogonal polarizations. The multimode fiber spectrometer is compact, lightweight, low cost, and provides high resolution with low loss.

1. Introduction

While traditional spectrometers are based on one-to-one spectral-to-spatial mapping, spectrometers can also operate on more complex spectral-to-spatial mapping [1–3]. In these implementations, a transmission matrix is used to store the spatial intensity profile generated by different input wavelengths. A reconstruction algorithm allows an arbitrary input spectrum to be recovered from the measured spatial intensity distribution. While this approach is more complicated than the traditional grating or prism based spectrometers, it affords more flexibility in the choice of dispersive element. For instance, spectrometers based on this approach have been demonstrated using a disordered photonic crystal [1], a random scattering medium [2], and an array of Bragg fibers [3]. We recently found that a multimode optical fiber is an ideal dispersive element for this type of spectrometer because the long propagation length and the minimal loss enables high spectral resolution and good sensitivity [4]. In a multimode fiber spectrometer, the interference between the guided modes creates a wavelength-dependent speckle pattern, providing the required spectral-to-spatial mapping. In the past, the contrast of this speckle pattern was found to depend on the spectral width and shape of the optical source [5–8], allowing researchers to use contrast as a measure of the laser linewidth [9]. As opposed to using only the statistical property of the speckle such as the contrast, we recently proposed and demonstrated that by recording the entire speckle patterns at different wavelengths, a multimode fiber can be used as a general purpose spectrometer [4]. The spectral resolution of the device depends on the spectral correlation width of the speckle, which is known to scale inversely with the length of the fiber [4,7,9]. The advantage of using an optical fiber is that a long propagation length is easily achieved with minimal loss, giving high spectral resolution. Furthermore, the fiber-based spectrometer requires only a multimode fiber and a monochrome CCD camera to record the speckle patterns. Compared to traditional spectrometers, optical fibers are lower cost, lighter weight, and can be coiled into a small volume while providing spectral resolution that is competitive with state-of-the-art gratingbased spectrometers. In this paper, we extend on the proof-of-concept demonstration presented in Ref [4]. And explore the operational limits of a multimode fiber spectrometer. We provide a theoretical analysis of the effects of the fiber geometry on the spectrometer performance, and then present a reconstruction algorithm combining a truncated inversion technique with a least squares minimization procedure, which enables accurate and robust spectral reconstruction in the presence of experimental noise. We also investigate the effects of spectral and spatial oversampling on the quality of the recovered spectra. Using a 20 meter long fiber, we are able to resolve two laser lines separated by merely 8 pm. A higher spectral resolution is expected for a longer fiber, but we are currently limited by the resolution of the tunable laser source used for calibration. We also discuss the bandwidth limitation when measuring a dense spectra due to speckle contrast reduction. To reduce the reconstruction error and increase the spectral bandwidth, we develop a method based on a polarization-resolved speckle measurement. Finally, we use a 2 centimeter long fiber to measure a continuous broadband spectrum generated by a supercontinuum source.

2. Operation principle of fiber spectrometer

The fiber-based spectrometer consists of a multimode fiber and a monochrome CCD camera that images the speckle pattern at the end of the fiber. The speckle pattern, created by interference among the guided modes in the fiber, is distinct for light at different wavelength, thus providing a fingerprint of the input wavelength. In our experiments, we used commercially available step-index multimode fibers with 105 μm diameter cores (NA = 0.22) and lengths varying from 2 cm to 20 m. A schematic of the experimental configuration is shown in Fig. 1(a). A near-IR tunable diode laser (HP 8168F) was used to provide a spectrally controlled input signal for the calibration and initial characterization. A polarization maintaining single-mode fiber was used to couple the laser emission into the multimode fiber through a standard FC/PC mating sleeve. The speckle pattern generated at the exit face of the multimode fiber was imaged onto an InGaAs camera (Xenics Xeva 1.7-320) with a 50 × near- IR microscope objective (NA = 0.55). The integration time of the camera was set to 100 μs. Alternatively, the speckle in the far-field zone may be projected directly onto the camera without the objective. Figure 1(b) is a movie showing the speckle patterns recorded at the end of a 20 m fiber as the input wavelength is scanned from 1500 nm to 1501 nm in the step of 0.01 nm. The speckle patterns decorrelate for very small changes in wavelength. Such high spatial-spectral diversity gives fine spectral resolution. We calculated the spectral correlation function of the speckle intensity, C(Δλ, x) = I (λ , x)I (λ + Δλ , x) / I (λ , x) I (λ + Δλ , x) −1, where I(λ, x) is the intensity at a position x for input wavelength λ, represents the average over λ. In Fig. 1(c), we plot the spectral correlation function averaged over many spatial positions across the 105 μm core of a 20 m fiber. The spectral correlation width, δλ is defined as C(δλ/2) = C(0)/2. When the input wavelength shifts by δλ, the output speckle pattern becomes nearly uncorrelated, C(Δλ = δλ) ≈0. In this case, δλ = 10 pm.

Fig. 1. (a) A schematic of the fiber spectrometer setup. A near-IR laser, wavelength tunable from 1435 nm to 1590 nm, is used for calibration and testing. Emission from the laser is coupled via a single-mode polarization-maintaining fiber (SMF) to the multimode fiber (MMF), with a standard FC/PC mating sleeve. A 50 × objective lens is used to image the speckle pattern generated at the end facet of the fiber to the monochrome CCD camera. (b) (Media 1) Movie showing the speckle pattern generated at the end of a 20 m multimode fiber as the input wavelength varies from 1500 nm to 1501 nm in the step of 0.01 nm. The wavelength is written on the top and also marked by the red line in the bottom scale. The speckle pattern decorrelates rapidly with wavelength, illustrating high spatial-spectral diversity. (c) Spectral correlation function of speckle intensity obtained from (b) exhibits a correlation width δλ = 0.01 nm, which enables fine spectral resolution.

To use the fiber as a spectrometer, speckle patterns such as the ones shown in Fig. 1(b) are recorded to construct the transmission matrix, as will be discussed in detail in section 4. After this calibration step, the tunable laser can be replaced by any optical source and the camera will record the speckle pattern. A reconstruction algorithm, to be discussed in section 4, is then applied to recover the spectra of the input. Note that in the experiments presented in this work, a single-mode polarization-maintaining fiber is always used to couple the signal to the multimode fiber. This ensures that the input to the multimode fiber will have the same spatial profile and polarization as the calibration. If the probe signal had a different profile or polarization, it could excite a different combination of fiber modes with different (relative) amplitudes and phases, making the calibration invalid.

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 Am and φm are the amplitude and initial phase of the mth guided mode which has the spatial profile ψm and propagation constant βm. 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.

4. Algorithm for spectral reconstruction

In the previous section, we investigated the effect of the fiber geometry on speckle decorrelation, and argued that the spectral correlation width determines the spectral resolution. More accurately, the spectral correlation provides a limit on the resolution, since we cannot distinguish between two wavelengths that produce highly correlated speckle patterns. In practice, due to the presence of experimental noise, the resolution of the fiberbased spectrometer also relies on the algorithm used to reconstruct the input spectrum from a measured speckle pattern. In this section, we describe a robust algorithm capable of reconstructing an arbitrary input spectrum in the presence of noise. Although the speckle patterns generated by different wavelengths could theoretically be calculated, as discussed in the previous section, this is not practical for a real fiber because it would require precise knowledge of the fiber geometry, including any twisting or bending, as well as the spatial profile of the input signal. Instead, we experimentally calibrated the transmission matrix for an input signal with a fixed spatial profile and polarization. The fiber was coiled and secured to an optics table before calibration and was not moved during subsequent testing. The monochrome CCD camera captures the intensity distribution across the exit face of the fiber, I (r) = S (λ )F (r,λ ) A(λ )dλ where S(λ) is the spectral flux density of the input signal, F(r, λ) is the position dependent transmission function of the fiber, A(λ) is the spectral sensitivity of the camera. Instead of measuring the spectral responses of the fiber and the camera separately, our calibration procedure measures the total transmission function of the fiber spectrometer, defined as T (r,λ ) = F (r,λ ) A(λ ) . It characterizes the spectral to spatial mapping from the input to the output of the fiber, as well as the spectral response of the camera:

(2)

In practice, the spectral signal is discretized into spectral channels centered at λi and spaced by dλ. If adjacent spectral channels are separated by more than the spectral correlation width δλ, they become independent because their speckle patterns are uncorrelated. Similarly, spatial discretization across the speckle image generates the spatial channels centered at rj, and they become independent if their spacing dr exceeds the spatial correlation length δr (equal to the average speckle size). In section 5, we will discuss this discretization process in more detail and explore the choice of the spatial and spectral channel spacing. In this section, we set dλ = δλ/2 and dr = δr. After discretizing the speckle patterns and the input spectrum, the transmission function T becomes a discrete matrix, and Eq. (2) becomes

I = T ⋅ S, (3)

where I is a vector representing the intensities in N spatial channels of the output, and S the intensities in M spectral channels from the input. Each column in the transmission matrix T describes the discretized speckle pattern, Ir, produced by incident light in one spectral channel. As an example, we consider the transmission matrix corresponding to a 1 m long fiber with spectral correlation width δλ = 0.4 nm. We calibrated a transmission matrix from λ = 1450 nm to 1550 nm in steps of 0.2 nm providing M = 500 spectral channels. From the spatial correlation function, we found that the speckle image contained ~600 spatial channels separated by δr, and we sampled 500 of these, N = 500. In section 5, we will discuss these choices in more detail. The transmission matrix was calibrated one column at a time by recording the speckle pattern generated at each sampled wavelength λi in S. The entire calibration process, consisting of recording speckle images at 500 input wavelengths, was completed in a few minutes. After calibration, we tested the spectrometer operation by measuring the speckle pattern produced by a probe signal and attempted to reconstruct the probe spectrum. An initial test was to recover a spectrum consisting of three narrow lines with unequal spacing and varying height. Since optical signals at different wavelengths do not interfere, we were able to synthesize the probe speckle pattern by adding weighted speckle patterns measured sequentially at the three probe wavelengths. In our first attempt to reconstruct the input spectrum, we simply multiplied the measured speckle pattern by the inverse of the transmission matrix: S = T −1 I. In Fig. 3(a), we plot the original spectrum with a red dotted line and the reconstructed spectrum with a blue solid line. This simple approach failed because the inversion of matrix T is ill-conditioned in the presence of experimental noise. There are several sources of noise in our spectrometer: (i) discrete intensity resolution of a CCD array, (ii) ambient light, (iii) mechanical instability of the fiber and fluctuation of experimental environment, (iv) intensity fluctuation and wavelength drift of the laser source. Among them we believe (iii) has the dominant contribution.

To understand why even a small amount of experimental noise can corrupt the reconstruction, we employ the singular value decomposition to factorize the transmission matrix, T = U D VT, where U is a N × Ν unitary matrix, D is a N × Μ diagonal matrix with positive real elements Djj = dj, known as the singular values of T, V is a M × Μ unitary matrix. The rows of V (resp. columns of U) are the input (resp. output) singular vectors and are noted Vj (resp. Uj). To find the inverse of T, we take the reciprocal of each diagonal element of D and then transpose it to obtain a diagonal matrix D’. The inverse of T is given as: T−1 = VD’UT. The issue we confront in the presence of noise is that the small elements of D are the elements most corrupted by experimental noise, and they are effectively amplified in D’ when we take their reciprocal. To overcome this issue, we adopted a “truncated inversion” technique, similar to the approach in Ref [3]. In this approach, we compute a truncated version of D’ in which we only take the reciprocal of the elements of D above a threshold value and set the remaining elements to zero. The truncated inverse of T is Ttrunc−1 = V D’trunc UT, which we use to reconstruct the input spectrum S = Ttrunc −1 I. Using this truncated inversion technique, we obtained a far superior reconstruction of the input spectrum, as shown in Fig. 3(b). Clearly the truncated inversion technique was able to recover the input spectrum. To provide a quantitative measure of the quality of spectral reconstruction, we calculated the spectrum reconstruction error, defined as the standard deviation between the probe spectrum and the reconstructed spectrum:

Note that in Ref [4], we used the signal-to-noise ratio (SNR) as a metric and showed that narrow lines were reconstructed with SNR greater than 1000. In this work, we chose to use the standard deviation because it allowed us to evaluate the ability of the spectrometer to reconstruct complex spectra. In order to optimize the truncated inversion, we reconstructed the input spectra using different threshold values for truncation and calculated the reconstruction error μ. The threshold of truncation was defined as a fraction of the largest element in D. For example, at a threshold of 0.01, any element in D with amplitude less than 1% of the maximal element in D would be discarded in the inversion process. In Fig. 3(c), we show the spectrum reconstruction error as a function of the truncation threshold. We found that the optimal threshold value was ~2 × 10−3 [this threshold value was used to reconstruct the spectrum shown in Fig. 3(b)]. After optimizing the truncation threshold, the singular value decomposition is only performed once for a given fiber, providing a Ttrunc −1 matrix which can then be used to recover any input spectrum with a single matrix multiplication.

The threshold is directly related to the experimental noise, when the noise increases, more singular values are perturbed and have to be discarded. One can understand how the optimal threshold behaves in the presence of noise by looking at its effect on the singular vectors of the T. Consider the perturbed matrix Tp = T + G, where G is a matrix of Gaussian noise with standard deviation σexp, representing the experimental error and T = U D VT is the transmission matrix of the system without noise. We can write TpVj = djUj + GVj, with dj the jth singular value of T and GVj a perturbation term. We have ||djUj|| = dj and GVj =σ exp M , where indicates averaging over realizations of the noise. We want to discard the singular values lower than the noise level, i.e. the singular values following exp d j <σ M. To obtain the threshold as defined previously, we have to compare this value to the largest singular value of T, which can easily be estimated when T is composed of independent elements, identically distributed with a mean value not equal to zero by d1 =τ NM , with τ the average value of the elements of T (i.e. the mean speckle intensity). We finally obtain an optimal threshold of the order of σexpN−1/2τ −1. In the reported experiment, we measured the noise variance by sequentially recording the matrix T twice for the same system, yielding σexp = 5 × 10−3. Based on this noise variance, we predicted an optimal threshold value of 1.2 × 10−3 in good agreement with the optimal value of 2 × 10−3 observed experimentally.

Fig. 3. (a) Initial attempt to reconstruct a probe spectrum, consisting of three narrow lines centered at 1484, 1500, and 1508 nm and with relative amplitude of 0.4, 1, and 0.6 (indicated by the red dotted line), simply by inversion of the transmission matrix: S = T−1 I fails, because this inversion process is ill-conditioned in the presence of experimental noise. (b) The same probe spectrum reconstructed using the truncated inversion technique described in the text. The truncation threshold was set to 2 × 10−3, and the spectrum reconstruction error μ = 0.028. The truncated inversion technique was able to recover the input spectrum, although background noise is still evident. (c) The spectrum reconstruction error, μ, of the reconstructed spectrum as a function of the truncation threshold. The minimal μ gives the optimal threshold value. (d) A further improved reconstruction was achieved using a nonlinear optimization procedure. The spectrum obtained from the truncated inversion technique was used as a starting guess to reduce the computation time, and the spectrum reconstruction error μ = 0.004.

In order to further improve the quality of spectral reconstruction, we also developed a nonlinear optimization algorithm. The spectral reconstruction process can be framed as an energy minimization problem, in which the optimal solution of the input spectrum, S, is the one that minimizes the energy E = ||I − T S||2 [1]. To find the optimal S, we developed a simulated annealing algorithm. At each step in the optimization routine, we changed one element in S at a time by multiplying it by a random number between 0.5 and 2. This provided us with a new spectrum, S’. We then calculated the change in energy ΔE = ||I − T S’||2 − ||I − T S||2 and kept the change (i.e. S = S’) with a probability equal to exp[−ΔE/T0], where T0 is the “temperature”. After performing this process once for every element in S, the temperature was reduced and the process was repeated. At high temperature the algorithm is more likely to accept “bad” choices (which increase the energy) in order to search broadly for a global minimum. Later in the algorithm, at lower temperatures, the algorithm is less likely to accept a “bad” choice. Note that changes which reduce the energy (ΔE < 0) are always accepted. The algorithm stopped after a fixed number of steps or if the energy dropped below a threshold value. In Fig. 3(d), we show the reconstructed spectrum obtained using the simulated annealing algorithm. The background noise is largely suppressed and the spectrum reconstruction error is reduced. The simulated annealing algorithm typically required a few hundred iterations to reach the optimal solution; however, the truncated inversion technique provides a good initial guess of S, which dramatically reduced the simulation time. In practice, these two reconstruction algorithms could be used in tandem. The truncated inversion technique provides a decent near-instantaneous reconstruction, which operates in real-time, since it requires a single matrix multiplication. The simulated annealing algorithm could be used to obtain a more accurate spectrum in applications where fast temporal response is less critical, or after the measurement is concluded. Of course, in both cases, the reconstruction will be improved by reducing the noise of the experimental measurements. In the current setup, the main source of noise is expected to be the stability of the multimode fiber, which was secured to an optical table during testing. Improved methods to rigidly stabilize the multimode fiber are expected to reduce the noise and provide more accurate spectral reconstructions. Using the reconstruction algorithm that combines truncated inversion and simulated annealing, we characterized the spectral resolution of the fiber spectrometer by testing its ability to discriminate between two closely spaced spectral lines. In order to synthesize the probe speckle pattern, we separately recorded speckle patterns at the two probe wavelengths and then added them in intensity. In Fig. 4, we show the reconstructed spectrum measured using a 20 m long fiber. The spectral positions of the probe lines (red dotted line) are in between the sampled wavelengths λi used in calibration. The reconstructed spectrum (blue solid line) clearly identifies the two peaks even though they are separated by merely 8 pm. Even higher spectral resolution is possible by using a longer fiber; however we would need a tunable laser with finer spectral resolution in order to calibrate such a fiber. This level of spectral resolution is competitive with the top commercially available grating based spectrometers.

Fig. 4. Reconstructed spectrum (blue solid line with circular dots marking the calibrated

wavelengths λi) of two spectral lines separated by 8 pm. It is obtained with a 20 m long fiber

spectrometer. The red dotted lines mark the center wavelengths of the input lines.

5. Spatial and spectral sampling

In contrast to the conventional spectrometers which map spectral information to one spatial dimension, our fiber spectrometer maps to two-dimensions (2D). This 2D spatial-spectral mapping fully utilizes the large detection area of modern 2D cameras to achieve large bandwidth of operation. The maximal number of independent spectral channels that can be measured in parallel is limited by the number of independent spatial channels that encode the spectral information. The number of independent spatial channels is equal to the number of speckles in the intensity distribution at the output end of the fiber, and it is also equal to the total number of guided modes, B, in the fiber if all modes are more or less equally excited. B is determined by the fiber core diameter, W, the NA, and the wavelength of light λ, B ≈π2 NA2W2 / 2λ2 [12]. The fibers considered in this work (W = 105 μm, NA = 0.22) support ~1000 modes at an operating wavelength of 1500 nm. In principle, increasing W or NA will increase the number of available channels for transmitting spectral information, thus increasing the bandwidth of the fiber spectrometer. In reality, however, the experimental noise may reduce the number of spectral channels that can be recovered simultaneously due to degradation of the spectral reconstruction. In this section, we investigate how the spectrum reconstruction error is affected by spectral and spatial sampling rates. We consider a 1 m long fiber with spectral correlation width δλ = 0.4 nm and a spatial correlation width δr = 4.1 μm. Our tunable laser covers the wavelength range from 1450 nm to 1550 nm, proving a maximum operating bandwidth of 100 nm. The speckle image had a radius of ~52.5 μm, providing ~600 spatial channels separated by δr. This number is less than the total number of guided modes in the fiber, because we did not excite all the modes [13]. Using this fiber, we generated various transmission matrices by adjusting the number of spectral and spatial channels, and the spacing of these channels. We then evaluated the reconstruction of a probe spectrum using different transmission matrices. The probe spectrum was a Lorentzian line centered at 1500 nm with a 1 nm full-width half-maximum (FWHM), synthesized by summing speckle patterns measured sequentially using the tunable laser, as discussed in the previous section. We used the truncated inversion algorithm to reconstruct the spectra and calculated the reconstruction error.

We first investigated the effect of the operating bandwidth on the ability of the

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