Radiometric errors in complex Fourier transform spectrometry

Abstract

A complex spectrum arises from the Fourier transform of an asymmetric interferogram. A rigorous derivation shows that the rms noise in the real part of that spectrum is indeed given by the commonly used relation σ_R = 2X ×NEP/(ηAΩ $\sqrt{\mathrm{\tau }N}$), where NEP is the delay-independent and uncorrelated detector noise-equivalent power per unit bandwidth, ±X is the delay range measured with N samples averaging for a time τ per sample, η is the system optical efficiency, and AΩ is the system throughput. A real spectrum produced by complex calibration with two complex reference spectra [Appl. Opt. 27, 3210 (1988)] has a variance σ_L ² = σ_R ² + σ_c ²(L _h - L _s)²/(L _h - L _c)² + σ_h ²(L _s - L _c)²/(L _h - L _c)², valid for σ_R, σ_c, and σ_h small compared with L _h - L _c, where L _s, L _h, and L _c are scene, hot reference, and cold reference spectra, respectively, and σ_c and σ_h are the respective combined uncertainties in knowledge and measurement of the hot and cold reference spectra.