Spectrum retrieval using circular shift procedure in complex Fourier-transform integrated-optic spatial heterodyne spectrometer

Abstract

Described is a noteworthy characteristic of a spatial heterodyne spectrometer featuring a complex Fourier transformation, namely that the spectrum can be retrieved by using a circular shift procedure when it is distributed over two adjacent spectral ranges. Retrieval is demonstrated by changing the temperature of the array and causing the spectrum to penetrate the adjacent spectral range.Introduction: A spatial heterodyne spectrometer (SHS) is an instrument that uses the Fourier transformation of a stationary interference pattern from a Mach-Zehnder interferometer (MZI) array [1]. The optical path differences between the two arms of the individual MZIs are designed to increase at equal intervals, which we denote as DL. Since the conventional SHS is based on the Fourier cosine transformation, the spectrum of the light launched into the MZI array should be finite only in the left half of the particular spectral range s/DL to (s + 1)/DL in wavenumber units, where s is an integer. When the spectrum is distributed either down to the Littrow wavenumber s/DL or over the centre (s + 1/2)/ DL, the spectral part outside the half range is folded over into the same range, resulting in the fatal deformation of the calculated waveform. This is the origin of the generation of blind regions where in practice we cannot measure the spectral components.We have proposed an advanced version of the SHS, namely the complex Fourier-transform integrated-optic SHS [2, 3], which generates in-phase and quadrature outputs at individual MZIs and acquires two interference patterns. This Letter reports a characteristic of our SHS, which is that the correct waveform of the spectrum can be retrieved even when it is distributed over two adjacent spectral ranges and therefore no blind regions are generated. The waveform is derived by circular shifting the Fourier transform when the spectrum is bandlimited to fall within the width 1/DL.