International Journal of Aerospace Engineering

Volume 2016 (2016), Article ID 2759121, 8 pages

http://dx.doi.org/10.1155/2016/2759121

## Simulation and Analysis of Spectral Response Function and Bandwidth of Spectrometer

^{1}Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun, Jilin 130033, China^{2}University of Chinese Academy of Sciences, Beijing 100049, China

Received 26 April 2016; Accepted 3 October 2016

Academic Editor: Andreas Ehn

Copyright © 2016 Zhenyu Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A simulation method for acquiring spectrometer’s Spectral Response Function (SRF) based on Huygens Point Spread Function (PSF) is suggested. Taking into account the effects of optical aberrations and diffraction, the method can obtain the fine SRF curve and corresponding spectral bandwidth at any nominal wavelength as early as in the design phase. A prism monochromator is proposed for illustrating the simulation procedure. For comparison, a geometrical ray-tracing method is also provided, with bandwidth deviations varying from 5% at 250 nm to 25% at 2400 nm. Further comparison with reported experiments shows that the areas of the SRF profiles agree to about 1%. However, the weak scattered background light on the level of 10^{−4} to 10^{−5} observed by experiment could not be covered by this simulation. This simulation method is a useful tool for forecasting the performance of an underdesigned spectrometer.

#### 1. Introduction

The spectral calibration [1, 2] of a spectrometer mainly consists of the accurate measurement of the Spectral Response Function [3, 4] (SRF, also known as Instrument Function [5]), from which the peak response wavelength and the spectral bandwidth of each detection channel can also be derived. Traditionally, experiments are usually performed to conduct the spectral calibration. Generally, a tunable monochromatic source such as a tunable laser [6] or a monochromator with higher resolution [7] is used to stimulate the spectrometer from wavelength to wavelength. For monochromators, the inverse process is also carried out, in the way of rotating the dispersion elements to scan a wavelength-fixed source. In addition, white-light interferometry [8], observing the spectral lines from a tungsten/discharge lamp [9, 10], or comparing reflectance of two different doped diffuse panels [11, 12] is also used by (imaging) spectrometers. The tunable wavelength scanning method theoretically can get all the SRF curves of each detection channel, but the operation is always so time consuming that only some typical positions or wavelengths can be measured in practice. Spectral line lamp and doped diffuse panel are only used for calibrating the peak response wavelength and are more effective for linear dispersion instruments. The white-light interferometry technology is not yet fully mature. Besides, experimental methods cannot be implemented until the instrument is assembled and adjusted. A simulation of SRF based on Line Spread Function (LSF) is reported by Mouroulis et al. [13]. The contour of the SRF is evaluated but it does not establish the relationship between the simulated SRF and the wavelength. Some parameters such as central wavelength and spectral bandwidth are still needed. Therefore, based on Huygens PSF, which is more accurate than Fast Fourier Transform (FFT) PSF, a modified simulation method for SRF and spectral bandwidth is suggested. Even in the optical design phase, the simulation procedure can also be carried out immediately at need to forecast the designer’s concerned specifications. This simulation method takes account of the effects of optical aberrations and diffraction, which are the two most important disturbances to SRF. The alignment and fabrication errors can be added as well for a postsimulation. In this paper, a spectrograph using array detectors is considered as a monochromator with multiple consecutive exit slits, and the detector characteristics are not considered here as they are relatively independent from the instrument’s intrinsic dispersion properties.

#### 2. Basic Principles in Spectral Response Function Simulation

The spectral integrated energy received by a spectrometer’s exit slit can be regarded as the slit’s response to the superposition of all the shifted monochromatic images of entrance slit [3, 13]. Considered in the wavelength domain, the received energy can be transformed to spectral density distribution versus wavelength, which is, namely, the SRF function of the exit slit. The SRF in fact is a spectral filer function of the spectrometer. Slit function can be simply assumed to be a rectangular one; image of entrance slit is the superposition integral of the entrance slit with the optical system Point Spread Function (PSF); if the optical system satisfies spatially invariant condition, the superposition integral can be replaced by convolution [14]. SRF of traditional spectrometer only focuses on the dispersion direction (assumed to be tangential direction), so the slit image will be obtained by using tangential Line Spread Function (). Typically, with the entrance slit having the same width as the exit slit, say, , the SRF of linear space invariant optical system can be expressed as where denotes cross-correlation and denotes convolution; is the tangential coordinate on image plane, which can be mapped onto wavelength domain coordinate by chief ray iterative tracings. Cross-correlation has an equivalent result on condition that the images of wavelengths near central are of the same profiles except being shifted. If is symmetric, the cross-correlation can be replaced by convolution. If is narrow compared with the slit width or ideally close to a Dirac delta function, then the SRF approximates to a triangle function with a bandwidth (FWHM, Full Width at Half Maximum) equivalent to in image plane coordinate. Otherwise, if the size is comparable with the exit slit, then it approaches a Gaussian function with a bandwidth broader than . If SRF can be approximated by a triangle or Gaussian function, all the spectral response characteristics can be identified by peak response wavelength and FWHM accurately.

The Huygens wavelets method and FFT are the common method to compute the diffraction PSF. Among them, the Huygens wavelets method imagines each point on a wavefront as a perfect point source with an amplitude and phase. Each of these point sources radiates a spherical “wavelet.” The diffraction intensity at any point on the image surface is the complex sum of all these wavelets, squared. The PSF is computed this way for every point on the image grid. So the Huygens PSF accounts for any local tilt in the image surface caused by the image surface slope, the chief ray incidence angle, or both. The optical system of a prism or grating spectrometer is not always axial symmetric as tilt surfaces often used, so using direct integration of Huygens wavelets method in place of FFT of the complex amplitude of wavefront at exit pupil has a higher accuracy in computing system PSF or . Generally, PSF is a nonelementary function with respect to either wavelength or spatial coordinates, so only the numerical expression form can be obtained through numerical analysis methods. A practical way to get the results is to extract them from optical design software (such as Zemax), where the calculation algorithm has been tested and verified in practice.

The numerical calculation procedure of SRF is based on (1). In order to establish the relationship between the contour of SRF and the corresponding wavelength, before the convolution, the coordinate system in wavelength space should be mapped onto a position coordinate system on image plane. It can be achieved by iteratively tracing chief rays on axis FOV (Field of View) at a serial of wavelengths. The wavelength corresponding to a chief ray intersected with the image plane at a certain position is referred to as the position’s* nominal wavelength*, while the position is called as the wavelength’s* nominal position*. This operation is performed only along the dispersion direction. For the wavelength , the in (1) can be calculated by integrating the discrete direction FOV Huygens . If the linear space invariant condition is satisfied, then only the zero FOV is needed. The monochromatic images of can be obtained by convoluting the with the rectangular object function. Then the calculation of SRF could be finished through a further convolution with the exit slit function. The SRF function corresponding to this exit slit could be obtained through inverse mapping the results from position coordinate system back onto the wavelength coordinate system.

The mutual mapping between wavelength and position coordinates is introduced to solve the nonlinear dispersion problem, as cross-correlation of exit slit with monochromatic images is carried out on a uniformly spaced position grid, while the SRF is a function of wavelength described in the wavelength domain. Calculation speed is the main drawback of Huygens PSF method. As the of specific wavelength changes continuously and smoothly with position coordinates on image plane, an algorithm of spline interpolating extracted of larger intervals will accelerate calculations effectively.

#### 3. Demonstration of Simulation Instance

A Féry prism monochromator [15] is proposed for illustrating the simulation algorithm. The monochromator adopts only one refractive-reflective prism with curved surfaces for light dispersion, while the wavelength scanning from 250 nm to 2500 nm is achieved via rotating the prism. The only exit slit is coplanar with the entrance slit and of the same size. Optical layout and construction parameters of the monochromator are showed in Figure 1 and Table 1, respectively. At each angle within the ±2.5° prism rotation range, the transverse aberration along dispersion direction is smaller than 8 micrometers. This optical system is used for solar spectral irradiance measurement [16].