Spectral noise due to measurement errors of Mach–Zehnder interferometer optical path phases in a complex Fourier-transform integrated-optic spatial heterodyne spectrometer Kazumasa Takada a,n , Mitsuyoshi Seino a , Akito Chiba a , Katsunari Okamoto b a Department of Electronic Engineering, Faculty of Engineering, Gunma University, 1-5-1, Tenjin, Kiryu, Gunma 376-8515, Japan b AiDi Corporation, 2-2-4 Takezono, Tsukuba, Ibaraki 305-0032, Japan article info Article history: Received 28 July 2012 Received in revised form 9 January 2013 Accepted 9 January 2013 Available online 11 February 2013 Keywords: Mach–Zehnder interferometer Optical waveguides Planar lightwave circuits Fourier transform spectrometer abstract We calculate the root-mean-square (rms) value of the spectral noise caused by optical path phase measurement errors in a spatial heterodyne spectrometer (SHS) featuring a complex Fourier transfor- mation. We show that the rms value is proportional to the rms error of the phase measurement and the proportionality coefficient is given analytically. The relationship enables us to estimate the possible performance of the SHS such as the sidelobe suppression ratio for a given measurement error. & 2013 Elsevier B.V. All rights reserved. 1. 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 planar waveguide version of the SHS architecture is a key approach since the MZI array is fabricated on one substrate. The actual optical path difference between the two arms of each MZI in a fabricated SHS is likely to deviate from the designed one, and the resultant deviated phase is estimated from the periodic change in the MZI output while sweeping the laser wavelength or heating either arm [2,3]. Since a spectrum is derived digitally by using the phase values measured at the individual MZIs, it is predicted that measurement errors would produce a spectral noise that degrades the performance of the SHS, for example the signal-to-noise ratio (SNR) of a finite spectrum component measurement and the sidelobe suppression ratio (SSR) of a sharp spectrum measurement. However, to our knowledge, no reports have described an explicit relationship between measurement error and spectral noise level. We have already proposed an advanced version of the SHS, namely the complex Fourier-transform integrated-optic SHS [4], which generates two different outputs from each MZI set in the in-phase and quadrature states. We report that the noise level obtained by applying the root-mean-square (rms) operation including arithmetic and ensemble averaging to the spectral noise distributed over the free spectral range (FSR) is proportional to the rms error of the phase measurement. This is derived by adopting a model of our SHS where the difference between the deviated phases of both states at each MZI is equal to p/2. The relationship enables us to estimate the potential performance of our SHS achieved when one of the most accurate methods is used to measure the phases. 2. Calculation The configuration of our planar waveguide SHS with an active phase shift scheme to produce the in-phase and quadrature states is shown in Fig. 1. There were 32 ( ¼ N) MZIs and the optical path differences between the two arms of the individual MZIs were designed to increase in equal increments of 240 mm, which is the product of the geometrical length difference and the waveguide effective refractive index. We assume that the spectrum to be measured falls within a particular FSR from h/DL ( ¼ s L ) to (h þ 1)/DL in wavenumber units, where h is an integer and s L is the Littrow wavenumber. We divide the FSR into M parts so that the digitized wavenumbers are s m ¼ s L þ m/(MDL) with m¼ 0, 1, 2, y, M 1 where the light intensities to be measured are {G m (t) }. We denote the output powers from the cross- and through-ports at the output coupler of the (k þ 1)th MZI as p k and q k as shown in the figure, where k is from 0 to N  1. We calculate the calibrated Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/optcom Optics Communications 0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.01.034 n Corresponding author: Tel./fax: þ81 277 30 1740. E-mail address: takada@el.gunma-u.ac.jp (K. Takada). Optics Communications 296 (2013) 61–64