Simulation Studies of an Ultraviolet Laser Absorption Imaging System
Introduction
To design practical engineering systems for modern energy and propulsion applications, such as rocket and jet propulsion systems, space reentry vehicles, and gas turbines, where extreme gas-dynamics often pose challenges on improving efficiency and ensuring system feasibility [1-3], it is critically important to inform practical designs with experimental insights of the thermochemical environment. Particularly, spatially resolved diagnostics of physical properties, such as temperature and concentrations of chemical species, are desired for the purpose of validating numerical simulations and chemical kinetic models [4]. In this context, an ultraviolet-laser-absorption-based imaging system becomes an attractive solution for providing 2D information on gas-dynamic environments of interest. An ultraviolet laser absorption imaging (UV-LAI) system is founded on the well-established principle of laser absorption spectroscopy (LAS), where magnitude of laser light attenuation at a certain wavelength is used to infer the concentration of absorbing chemical species [5]. The light attenuation arise from the resonance interaction between light and matter, which sees a molecule transitioning to a higher energy state when excited by light carrying energy that matches the energy gap of the transition. A fixed-wavelength, continuous-wave laser emitting in the UV region has the unique advantage of accessing strong electronic energy transitions of diatomic species, such as hydroxyl radical (OH) and nitric oxide (NO), important for high-temperature combustion and air chemistry [6]. This enables high signal-to-noise ratio (SNR) in the measured light attenuation, and therefore, improves the accuracy of the species concentration measurements. However, to enable accurate 2D measurements using the UV-LAI system, the impact of optical aberration and light refraction through the test gas on the interpretation of the final image needs to be well understood.
In this work, various simulation tools were utilized to perform system-wide simulations for an UV-LAI concept. The imaging system of interest, shown in Fig. 1-1, consists of a 226-nm (suitable for probing NO), continuous wave laser source, which is expanded by a beam expander before being passed through an absorbing test gas. The transmitted beam is then collected by a lens onto a camera sensor. Two types of optical design software, deploying sequential and non-sequential ray tracing, were explored, and the relative advantages of each type of ray tracing strategy were discussed. As a first attempt, Blender and PBRT were used to simulate the image response of a simple geometry consisting of two overlapping spherical bodies. A collimated source and a point source configuration were used. Then, Zemax was deployed to investigate the optical aberration of a beam expander. The real-lens effect of practical beam expander system was compared with an ideal paraxial system. LightTools was adopted to simulate the sensor illuminance resulted from the UV-LAI system. A spherical body was first used in a simplified source-gas-sensor configuration absent of a imaging lens and a beam expander. The impact of refraction and source light divergence angle on the final absorption image was investigated via parametric studies, which led to the optical design of a simple correction lens scheme to reduce measurement error. Finally, the image response from the whole UV-LAI system was simulated in LightTools with the Zemax-designed beam expander, and the result showed minimal impact of the real beam expander compared to a ideal paraxial system.
Background
Ultraviolet Laser Absorption Imaging
The UV-LAI system measures the attenuation of light through an absorbing medium deploying the same fundamental principle of LAS, which is illustrated in Fig. 2-1. Light attenuation at a certain wavelength Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda } is evaluated by absorbance (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} ), which is defined as follows:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = -\ln{\left(\frac{I_t}{I_o}\right)}_\lambda = = -\ln{\left(\frac{Absorption\ image}{Background\ image}\right)}_\lambda\ (\mathrm{Eqn.}\ 2-1), }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_o } is the background intensity absent of absorption, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_t } is the transmitted intensity with absorption. In an imaging setup, a background image is recorded first, and then a separate absorption image is recorded by introducing the test gas. The signal recorded on each pixel of the background and absorption images was plugged into Eqn. 2-1 to compute the absorbance distribution.
To evaluate the error in the 2D absorbance measurement caused by optical aberration and refraction through the test gas in system-wide simulations, a ground truth reference is first produced by setting the refractive index of the test gas to be the same as that of the ambient gas in a perfectly collimated system. The background images were obtained by setting the optical density (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D } ) of the test gas to zero. The optical density is defined as:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D = -\log_{10}{\left(\frac{I_t}{I_o}\right)\frac{1}{L}}\ (\mathrm{Eqn.}\ 2-2). }
When simulating the absorbance response with refraction through the test gas, the test gas' refractive index is set to a value smaller than the ambient gas (representative of a high-temperature reacting gas-dynamic environment), and the background and absorption images were obtained by changing the test gas' optical density. Note that this approach still preserves the refraction of light through the test gas in the background image, allowing the direct impact of refraction to be studied through simulations. To achieve the equivalent effect in future practical experiments, the laser wavelength could be shifted slightly such that the test gas stops absorbing light.
With the absorbance signal measured from the background and absorption images, species mole fraction can be obtained via the Beer-Lambert relation [5]:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = n_{tot} \chi_i \sigma_\lambda L\ (\mathrm{Eqn.}\ 2-3), }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_{tot}} is the total number density of the gas which can be calculated using ideal gas law in practical experiments; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_i} is the mole fraction of species Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_\lambda} is the absorption cross section of species Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} at wavelength Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} and is obtained from a spectroscopic model of species Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} in practical experiments; </math>L</math> is the absorption path length, which measures the distance that light travels within the absorbing test gas. In this work, only absorbance images were simulated and no additional steps were taken to calculate the species mole fraction.
Methods
Sequential vs. Non-Sequential Ray Tracing
| Sequential | Non-Sequential | |
|---|---|---|
| Description | Rays are propagated sequentially from one object to the next. | No pre-defined sequence of surfaces which rays that are being traced must hit |
| Strength | Cleaner model, easier to simplify and optimize. | More flexible to model complex light system. |
| Weakness | Less Flexible | Requires accurate 3D models, more difficult to set up |
| Application | Imaging and afocal systems, lens design | Analyze optical systems with stray light, scattering and illumination. |
| Tools | Zemax, Code V | LightTools, ISET 3D |
Results
PBRT and Blender
Zemax Opticstudio
The modeling procedure includes system setup, optic surface setup, and optimization. Starting from the system setup shwon in Fig. 4-3, where the aperture value is set to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 mm } with uniform distribution, and the wavelength is set to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.226 \mu m } , which matches the property of the input beam. The system is defined as an "Afocal" image space since the input and output should all be collimated, meaning the system has no focus, or infinite focal length.
Firstly, we can start with modeling the system with all paraxial lenses. Seven optical surfaces are built in the model, where surface 0 and 1 define the input collimated beam with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.5mm } semi-diameter; surface 2 is the first paraxial lens that bend the beam for expanding, surface 3 and 4 control the distance to the collimating lens at surface 5 with the target semi-diameter of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 50.8mm } ; the sensor is placed at surface 6 for evaluation. The designed system is shown in Fig. 4-4.
In achieving the desired beam size and collimation level, the design optimizer provided by Zemax Opticstudio is utilized, as shown in Fig 4-5. Where we set the constraints to help calculating the lens properties in order to achieve desired collimated output beam with reasonable system configurations. First, we set the target output beam diameter of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 50.8mm } using the function "REAY", and assigned a higher optimization weight of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10 } . Then we constrained the total length from the expanding lens to the collimating lens in between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 } to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 406mm } , also with a higher optimization weight of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10 } . Finally, we defined "EFLX" parameters to monitor the focal lengths of the two individual lens at surface 2 and 5, and the focal length the entire system.
LightTools and CODE V
To simulate the illuminance on the camera sensor in a UV-LAI system, LightTools' forward Monte Carlo ray tracing module was deployed to first model a simplified imaging setup with a collimated source, a spherical test gas, and a sensor. CODE V was then used to aid the design of a simple correction lens to counter the refraction caused by the test gas. Finally, LightTools was again utilized to model the whole UV-LAI system including the beam expander designed in Zemax (see the section on Zemax) and the correction lens.
Figure 6-1 shows the setup geometry of the simplified UV-LAI system in LightTools. The light source was modeled as monochromatic at 226 nm with a uniform power distribution. The optical property of the test gas, modeled as a uniform spherical body, was set to be 100% transmitting neglecting reflection of light at the gas surface. A user-defined material was used for the test gas with specified index of refraction and optical density. 1 million rays were used in all forward Monte Carlo ray tracing simulations, and 250 x 250 grid divisions were used for the sensor ("receiver" in LightTools' notation) to obtain illuminance distributions. Sample background and absorption images obtained from this simple setup is shown in Fig. 6-2. The first row of background and absorption images in Fig. 6-2 is the result when refraction through the test gas is considered, with the ratio of refractive index ratio crossing from ambient to the test gas, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_2/n_1} , defined as 0.98. The optical density, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} , was chosen to be 0.02/mm. The second row of images in Fig. 6-2 shows the results when refraction is turned off (setting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_2/n_1} =1), representing the ground truth level of illuminance as discussed in the background section.
To investigate the impact of different levels of refraction through the test gas, a parametric study of the change in index of refraction from the ambient gas to the test gas (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_2/n_1} from 0.96 to 0.99) was performed, and the absorption image results are shown in Fig. 6-3. Sharper edge spread of illuminance profiles across the sphere can be observed in Fig. 6-3 with lower Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_2/n_1} . The absorbance profiles across the center of the sphere (sampling along y = 0 mm) were computed from the simulations and compared to the ground truth level in Fig. 6-4. Significant error in the absorbance profile (>100% when close to the edge of the sphere) can be observed when refraction is introduced. Largest error was seen to occur at the edge of the sphere, and immediately outside of the sphere, some absorbance was also found. As in the unbiased scenario, outside of the sphere, light should not be absorbed, the absorbance observed was a artifact of refraction through the test gas. Since the index of refraction reduced across the test gas, light is steered away from the center of the sphere, leading to light that has traversed inside of the test gas to fall outside of the apparent edge of the sphere in the final image. Interestingly, from the parametric study it can be seen that the increase of relative error is the largest when changing from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_2/n_1} = 1 to 0.99, suggesting that even a small change can have strong contribution to errors during the interpretation of absorbance.
To explore the effect of ray divergence angle (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} ) on the interpretation of absorbance distributions, a 3Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^\circ} divergence was introduced in the light source in LightTools, and the resulting absorbance distributions of the ground truth, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} = 0, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} = 3Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^\circ} cases are shown in Fig. 6-5. The location of the test gas sphere's edge can clearly be seen in Fig. 6-5 to broaden when a divergence angle is introduced, suggesting in practical UV-LAI implementations, distortion correction targets are likely necessary to ensure the spatial scaling from pixels to physical distances can be correctly obtained.
With the simplified UV-LAI system simulated with LightTools, the possibility of a simple correction lens scheme was explored in COVE V. The design strategy of the correction lens is that the lens would have opposite power to that of the test gas and the image scale would be maintain through a unity magnification after the lens. The optical setup calculated in CODE V is shown in Fig. 6-6. For simplicity and potential future implementability, a 3-inch-diameter stock lens from Thorlabs (LA4795) was used for the sequential ray tracing in CODE V. The lens focal length at 226~nm was computed as 168.5 mm in CODE V, suggesting image and object distances of 337 mm for the correction lens scheme.
The correction lens was then modeled in LightToools as shown in Fig.6-7. To examine the improvement in the illumination uniformity after the introduction of the correction lens, as a completely uniform background should be resulted if the refraction through the test gas is fully corrected, background images from the cases with and without correction lens and the ideal reference are shown in Fig. 6-8.
While effect of refraction is still present, as signified by the contrasting illuminance level at the edge of the test gas in Fig. 6-8, the level of contrast caused by refraction through the test gas is clearly reduced compared to the case without the correction lens. The reduction of absorbance error is further demonstrated through comparing the absorbance profile obtained with the correction lens to that of the uncorrected simulations, as shown in Fig. 6-9 (left). To investigate if the improvement in accuracy can be retained for more complex, 3D geometries, the 2-sphere setup described in the section on PBRT is simulated with the correction lens, and the absorbance profiles are shown in Fig. 6-9 (right). It can be seen that the correction lens provides good improvement for both the simple sphere and the more complex 2-sphere geoemtries.
Finally, to explore the effect of real beam expander on the interpretation of the final results, the beam expander described in the section on Zemax is modeled in LightTools with the correction lens. The LightTools setup is illustrated in Fig. 6-10. The 2-sphere geometry was used in the simulations.
The comparison of background and absorption images of the system with and without the real beam expander is shown in Fig. 6-11. At the center of the larger sphere, both the background and absorption images produced by the system with the real beam expander shows higher intensity when compared with the results obtained with a perfectly collimated source, a result caused by aberration of the real beam expander. However, from the computed absorbance traces sampled across the centers of the 2 spheres, as shown in Fig. 6-12, no significant difference was observed between the case with a perfectly collimated source and a real beam expander. This is due to the fact that the calculation of absorbance takes the ratio between the absorption image and the background image. Therefore, since the higher intensity at the center of the larger sphere was found in both images in the real beam expander case, the effect is canceled out when taking the fraction. The close agreement in the perfect collimation and real beam expander cases in Fig. 6-12 also suggests a more forgiving tolerance on the beam expander aberration for a UV-LAI system.
Conclusions
In this work, a ultraviolet laser absorption imaging (UV-LAI) system was simulated using various sequential (Zemax and CODE V) and non-sequential (Blender, PBRT, and LightTools) ray tracing tools. It was found that sequential ray tracing software is most useful for designing optical setup with the specific goal of quantifying and minimizing aberrations, while non-sequential ray tracing tools can be used to model the illumination of a entire imaging system.
From the parametric study on refractive index change across the surface of the test gas, it was found that refraction through the test gas can introduce significant error in the spatial distribution of absorbance, which is detrimental for measuring physical properties through the UV-LAI system. Interestingly, among the levels of refractive index change studied, it was found that changing from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_2/n_1} of 1 to 0.99 gives the largest rise in relative absorbance error. The effect of the divergence angle of the light source was also investigated, and distortion in the spatial scale of image features (e.g., edge location) was observed when a 3Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^\circ} divergence in the light source was introduced. This suggests in practice implementations of the UV-LAI system, where the divergence angle of the light source could not be guaranteed to be negligible, distortion targets (e.g., a dot array target) are likely necessary to ensure the correct pixel-to-physical-distance scale can be recovered.
A simple correction lens scheme was designed and resulted in notable improvement in the accuracy of the image-measured absorbance distribution. However, large error was nevertheless found at the edge of the test gas. Finally, the effect of a real beam expander on the final spatial absorbance profile was explored in simulations. The results suggest that while aberration effects were observed in the background and absorption images, after computing the absorbance, those effects are not significant.
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