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 ≈π2 NA2W2 / 2λ2 [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