2.1

Optical Design

The instrument is a passive spectrometer operating in the Short Wavelength Infrared (SWIR) part of the spectrum from 1.588 µm to 1.673 µm. It is intended to conduct the first atmospheric reconnaissance from a UAV platform and be tested to qualify for a future space mission. The instrument specifications are listed in Table 1.

The optical layout of the instrument was designed using OpticStudio^® ZEMAX software which enables optical simulation. The Instrument optical design mainly consists of five optical elements, slit, collimator, diffraction grating, camera (focusing unit), and detector. Figure 1 shows the instrument’s preliminary optical layout produced by Zemax.

Table 1:

Instrument parameters

Figure 1:

Instrument Optical Layout. Colors represent the wavelengths where Blue is the minimum of 1.588 µm, Green is the central wavelength 1.6305 µm, and Red is the maximum wavelength of 1.673 µm.

2.1.1

Input Slit

Etendue is a significant quantity that defines the amount of light collected by the instrument, and it must be matched for all the optical components in the instrument¹⁴. The following equation describes it as:

where A is the area of the slit surface and Ω is the solid angle at which light is accepted into the instrument.

It is acknowledged that there must be a trade-off between the slit size and the spectral resolution of the instrument to maintain a high signal-to-noise ratio (SNR) with acceptable spectral resolution.

In this instrument, light is collected by a conventional field lens, which must have an F/# of 3.3, and will pass through a 25 μm slit width that has a length of 3 mm. Figure 2 shows the slit simulation in Zemax. The slit is positioned perpendicular on the transmission grating to enable Littrow condition with an angle of 29°.

Figure 2:

A close-up look at the slit. Slit length is 3 mm, while the width is 25 μm (0.025 mm). The rectangular slit is simulated to model the light at three different positions in the rectangular. On-axes positions are shown in the middle at three points, one at the centre and two at the edges. Off-axes positions are shown at the two extreme ends of the rectangular slit.

2.1.2

The Collimator, Grating, and Camera

We have experimented with a wide variety of lenses available in Zemax libraries from different manufacturers, e.g., ThorLabs, Edmund Optics, and Newport, and found preliminary suitable optical components for the system. Starting with the collimator, we have selected an achromatic lens coated for the Near Infrared (NIR) and Short Wavelength Infrared (SWIR) bands from Edmund Optics^a. It has a diameter of 30 mm, an effective focal length of 100 mm, and thus an F/# of approximately 3.33. It is shown in Figure 1 as a white rectangle with no further details because the manufacturer only provided a ZEMAX Blackbox of this lens to preserve their intellectual property rights. The collimated light will illuminate a transmission grating (Wasatch Photonics^b) with a 600 line/mm with a diameter of 50.8 mm, which is oriented in a Littrow configuration with an incident angle of 29°. The diffraction grating offers an average efficiency of up to 90% at 1.588 μm. It, however, drops to approximately 86% at 1.673 μm, as shown in Figure 3. Diffracted light will then be brought into a focal point at the detector using two achromatic lenses (Edmund Optics) with a diameter of 25 mm. The camera lenses have focal lengths of 200mm and 150mm, respectively, and are separated by a distance of about 108.5 mm resulting in a total effective focal length of 47.9 mm.

Figure 3:

Typical transmission grating efficiency specified by Wasatch Photonics The efficiency of S- and P-type polarized light are shown in red and blue lines, respectively. The average efficiency is shown in the green line. Figure credit: © 2021 Wasatch Photonics.

2.1.3

Detector

An Indium Gallium Arsenide (InGaAs) based detector is a perfect candidate in the SWIR range for its compactness, high Quantum Efficiency (QE)¹⁵ and being the most cost-effective option we have. We aim to employ a high-speed, 640 × 512 pixels detector with a QE of approximately 70% from the First Light Imaging. It has a length and width of 55 mm and a pixel pitch of 15μm resulting in an active area that fully accommodates all wavelengths, as shown in Figure 4.

Figure 4:

Footprint diagram of the image on the detector. The aperture’s full Y height is 9.6mm, representing the 640 pixels. The X width of the aperture (7.6mm) represents the 512 pixels on the detector.

3.1

ZEMAX Simulation

The slit is modelled by selecting nine positions (fields), 3 in the middle, 3 in the right, and 3 in the left, as shown in Figure 2. The initial system performance was tested by ray tracing and generating the Spot Diagram of each field position for a range of wavelength, as shown in Figure 6. The columns represent the minimum, central and maximum wavelengths with a spectral step (resolution) of 0.4 nm. The system’s minimum wavelength, 1588 nm, is shown in group A and advanced with a spectral step of 0.4 nm twice, giving 1588.4 nm and 1588.8 nm shown in the second and third columns, respectively. The central and maximum wavelengths, 1630.5 nm and 1673 nm are shown in group B and C, respectively, with the same spectral step. The filed positions on the slit are shown in the rows. Group 1 has three positions at the center of the slit. Group 2 and 3 show another 3 points at the right and left edges of the slit, respectively. The spot diagram shows that all rays are confined within the Airy disk with a radius of 12.14 μm. It is also noteworthy that the spot diagram illustrates a relatively consistent and uniform performance of the system on and off-axis.

Figure 5:

Matrix Spot Diagram of filed positions in slit shown in Figure 2. Columns show the wavelengths 1588 nm, 1630.5 nm, and 1673 nm with a spectral step of 0.4 nm. Rows represent positions selected on the slit center and the two extreme edges. The box size is 30 μm, and the Numerical Aperture (NA) is 0.1.

Figure 6:

Central wavelength 1630.5 nm with two nearby wavelengths, 1630.9 nm, and 1631.3 nm, separated by 0.4 nm.

The system’s spectral resolution was tested in ZEMAX by generating the Geometric Image Analysis (GIA) feature, which we use to show the wavelengths separated by the required spectral step can be resolved. Figure 7 shows the central wavelength of 1630.5 nm with two adjacent wavelengths, 1630.9 nm and 1631.3 nm, separated by 0.4 nm spectral step. The normalized radiance profiles of these wavelengths’ peaks are shown in Figure 8. The Full-Width-at-Half-Maximum (FWHM) value in figure 8 indicates that the slit image is sampled by at least 2 pixels on the detector.

Figure 7:

Normalized Radiance profile of the wavelengths peaks in figure 6

Figure 8:

Instrument spectral interval generated by NASA’s Planetary Spectrum Generator (PSG). The lines resolution is set to 0.4 nm to simulate the instrument spectral resolution.

4.1

Synthetic Spectral Responsivity

The instrument operates in the SWIR interval 1.588 μm – 1.673 μm, which includes some atmospheric molecular absorption bands, mainly CO₂, CH₄ and H₂O. This spectral domain offers minimal interference by H₂O and relatively high signal-to-noise ratio (SNR) compared to thermal channels. The solar radiance that reflects off the ground and travels back to space carries various information about the atmosphere that can be inferred, i.e., gas concentration, temperature, and pressure. The Planetary Spectrum Generator (PSG)¹⁶ is an online accurate radiative transfer code developed to solve the entire atmosphere’s radiative transfer equation. It allows the user to enter all the parameters related to Earth, atmosphere, and instrument. Thus, it has been used to simulate our instrument’s spectral range, where data is stored and processed in MATLAB. Figure 9 shows the instrument spectral range that includes absorption bands of CO₂ and CH₄. Figure 10 shows the gas transmittance in the instrument spectral window. The main atmospheric and spectroscopic parameters used in this Line-by-Line calculation are listed in Table 2.

Figure 9:

Gases transmission in the atmosphere. CO₂ (red) CH₄ (blue), and H₂O (green), profiles are plotted with standard concentrations in the atmosphere.

Table 2.

Calculation parameters used to solve the Radiative Transfer Equation in the synthetic atmosphere.

4.2

Retrieval Algorithm

The challenges of retrieving atmospheric constituents from space & aircraft (including UAVs)-based platforms are well known to the Atmospheric and Space physics community. However, much effort has been put into developing retrieval algorithms to meet various mission’s needs^17-22. These algorithms share a basic fundamental approach that requires forward modelling, but each employs different inversion techniques to infer the atmospheric parameters of interest. The inversion algorithms offer a significant mean to infer specific components from the radiative transfer equation, which is not invertible analytically because of its non-linearity nature.

The adapted retrieval method for the instrument is similar, in concept, to that of OCO-2¹⁸, GOSAT²¹, and SCIAMACHY^19,20 missions retrievals where the algorithm starts with a forward model that includes instrument parameters to generate a synthetic measurement and ends by employing an inversion method, Optimal Estimation²³, to infer the atmospheric parameters of interest. Furthermore, the Levenberg–Marquardt inversion algorithm²⁴ will be evaluated once the flight data is ready for analysis. The forward model can be mathematically represented by the measurement vector, y as follows:

Equation 2 states that the spectrum (e.g., measurement) y is a function (F)of the parameters, b for the instrument parameters, c for the known physical parameters, and x for is the state vector of the target (e.g., the atmospheric gas of interest). The ɛ term represents the noise or random error in the set of measurements²⁵.

As mentioned previously, to inverse the equation, an iterative retrieval algorithm based on Bayesian optimal estimation²³ will be employed. Bayes’s rule finds the probability of a specific event by combining a priori knowledge about the atmospheric state to obtain the posterior state. Bayes’s theorem can be implemented, assuming Gaussian statistical distribution, in the retrieval algorithm and the maximum posterior probability state and its covariance matrix Ŝ are:

where the x and y are the state vector and measured spectrum, respectively, illustrated in Eq. (2), S_ε is the measured error covariance, x_a and S_a are the priori state vector and priori covariance matrix, T is the transpose operator, K is the weighting function¹⁸ and F(X) is the forward model expressed in Eq. (2).