View source for 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.

[[File:UV-LAI setup.png|450px|thumb|left|Fig. 1-1: Schematic of the UV-LAI system targeted for simulation study in the present work.]]

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 === 

[[File:absorbance.png|450px|thumb|left|Fig. 2-1: Schematic of a basic LAS setup.]]

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  <math> \lambda </math>  is evaluated by absorbance (<math> \alpha</math>), which is defined as follows:

<math> \alpha = -\ln{\left(\frac{I_t}{I_o}\right)}_\lambda = = -\ln{\left(\frac{Absorption\ image}{Background\ image}\right)}_\lambda\ (\mathrm{Eqn.}\ 2-1), </math>

where <math> I_o </math> is the background intensity absent of absorption, and <math> I_t </math> 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 (<math> D </math>) of the test gas to zero. The optical density is defined as:

<math> D = -\log_{10}{\left(\frac{I_t}{I_o}\right)\frac{1}{L}}\ (\mathrm{Eqn.}\ 2-2). </math>

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]:

<math> \alpha = n_{tot} \chi_i \sigma_\lambda L\ (\mathrm{Eqn.}\ 2-3), </math>

where <math>n_{tot}</math> is the total number density of the gas which can be calculated using ideal gas law in practical experiments; <math>\chi_i</math> is the mole fraction of species <math>i</math>, <math>\sigma_\lambda</math> is the absorption cross section of species <math>i</math> at wavelength <math>\lambda</math> and is obtained from a spectroscopic model of species <math>i</math> 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 and non-sequential ray tracing are two approaches used in optical design and analysis to simulate light rays as they propagate through optical systems. (See TABEL 3-1)

In sequential ray tracing, light rays are traced through a specific order from the entrance pupil to the exit pupil. This approach is well-suited for systems with well-defined optical paths, such as cameras, telescopes, and microscopes. Sequential ray tracing is computationally efficient and provides accurate results for systems where light travels along a single, predictable path. (Fig. 3-1 (A))

Non-sequential ray tracing, on the other hand, allows for the simulation of complex optical systems where light rays do not follow a predetermined path. This is particularly useful in the analysis of systems with multiple interacting components, such as scattering elements, diffusers, or light guides. Non-sequential ray tracing is employed when the order of optical elements is not fixed, and interactions between rays and surfaces are more dynamic. This method is essential for modeling systems like illumination systems, sensors, and systems with free-form optics. (Fig. 3-1 (B))

In this project, we've examined several different optic molding tools for varies aspect of our optic systems. In the end, we've found that Zemax and Code V are suitable tools for sequential molding, while the LightTools and ISET 3D are powerful tools for non-sequential modeling. 

{| class="wikitable" style="margin:auto; background-color:#ede7e7"
|+ TABLE 3-1 Comparison between Sequential and Non-Sequential Modeling Methdos
|-
!  style="background-color:#8d3c3c; color: #ffffff"| !! style="background-color:#8d3c3c; color: #ffffff""| Sequential !!style="background-color:#8d3c3c; color: #ffffff""|  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
|}



[[File:Figure 3-1.gif|500px|thumb|centre|Fig. 3-1: Illustration of the difference between sequential raytracing and non-sequential raytracing. (A) In sequential raytracing, there is a one-to-one relation between object points and image points. (B) In non-sequential raytracing, no image will be formed. [7]
]]

== Results ==

=== PBRT and Blender ===

As a first step towards simulating our model, we investigated the abilities of Blender, PBRT, and ISET3D to render a simplistic two-sphere model in different light sources. 

With Blender, we began by creating the geometry of the two spheres. We used Blender's native rendering tools, testing multiple built-in materials settings, and the ways that they responded to different light sources (collimated, point, and beam). Figure 4-1 shows the Blender renderings of the model under two interesting light sources (collimated and point source), as well as in two different materials (diffuse and translucent). 

[[File:blender.png|600px|thumb|centre|Fig. 4-1: Blender rendering of two-sphere model in different light sources]]

In order to explore more quantitative evaluation of our model, we then attempted to import the Blender-created geometry into PBRT and ISET3D. Using PBRT, we were able to render the two spheres under a variety of different illuminations and material qualities (two examples of which are shown in Figure 4-2). Unfortunately, due to technical difficulties connecting to a remote system, we were unable to fully explore the capabilities of ISET3D. 

[[File:pbrt.png|300px|thumb|centre|Fig. 4-2: PBRT rendering of two-sphere model in different light sources]]

The main utility of PBRT and Blender would be in handling more complex geometries. Figure 4-3 shows an experiment we carried out, in which a more complex dinosaur mesh model was constructed in Blender and rendered in PBRT. This would be critical in instances if our sample was very oddly shaped, or its shape played a key role in the measurements. However, for the purposes of this study, our measured sample was being modeled as a very simple spherical geometry, since the primary focus of our simulation was to examine the behavior of light through the imaging apparatus. As a result, we began investigating other tools as well to better model the full system setup.

[[File:pbrt-dino.png|200px|thumb|centre|Fig. 4-3: PBRT rendering of more complex dinosaur mesh]]

=== Zemax Opticstudio ===

In order to build a model for optimizing the beam expander, we need to firstly define the material properties of the lens. These properties are typically represented by glass dispersion formulas. Since the relationship between the refractive index of different media and the wavelengths of rays is usually non-linear, various approximation formulas are employed for different classes of materials and different wavelength ranges.
For example, the widely used Schott Formula:

<math> n^2 = a_0^2+a_1\lambda^2+a_2\lambda^{-2}+a_3\lambda^{-4}+a_4\lambda^{-6}+a_5\lambda^{-8}   (\mathrm{Eqn.}\ 5-1), </math>

the Herzberger Formula that is used in the infrared spectrum:

<math> n(\lambda) = A+BL+CL^2+D\lambda^2+E\lambda^4+F\lambda^6     (\mathrm{Eqn.}\ 5-2), </math>

where,

<math> L = \frac{1}{\lambda^2 - 0.28} </math>

or, the Conrady Formula that is mostly useful in fitting sparse data:

<math> n =n_0 + \frac{A}{\lambda} + \frac{B}{\lambda^{3.5}}   (\mathrm{Eqn.}\ 5-3). </math>

In this study, we use the very common Sellmeier equation for the SILICA material:

<math> 
n^2 -1 = \frac{K_1\lambda^2}{\lambda^2-L_1} + \frac{K_2\lambda^2}{\lambda^2-L_2} + \frac{K_3\lambda^2}{\lambda^2-L_3}   (\mathrm{Eqn.}\ 5-4). 
</math>

The material setup is shown in Fig. 5-1, and the refractive index plot is shown in Fig. 5-2. The refractive index for 226nm ray used in this project is <math> 1.5233 </math>.

[[File:Figure 4-1.png|500px|thumb|centre|Fig. 5-1: Material Selection: SILICA]]


[[File:Figure 4-2.jpg|500px|thumb|centre|Fig. 5-2: SILICA Refractive Index Plot]]


The modeling procedure includes system setup, optic surface setup, and optimization. Starting from the system setup shwon in Fig. 5-3, where the aperture value is set to <math> 1 mm </math> with uniform distribution, and the wavelength is set to <math> 0.226 \mu m </math> , 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.


[[File:Figure 4-3.png|200px|thumb|centre|Fig. 5-3: Afocal System Setup]]

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 <math> 0.5mm </math> 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 <math> 50.8mm </math>; the sensor is placed at surface 6 for evaluation. The designed system is shown in Fig. 5-4.

[[File:Figure 4-4.png|500px|thumb|centre|Fig. 5-4: Paraxil Lens System Design]]

In achieving the desired beam size and collimation level, the design optimizer provided by Zemax Opticstudio is utilized, as shown in Fig 5-5a. 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 <math> 50.8mm </math> using the function "REAY", and assigned a higher optimization weight of <math> 10 </math>. Then we constrained the total length from the expanding lens to the collimating lens in between <math> 0 </math> to <math> 406mm </math>, also with a higher optimization weight of <math> 10 </math>. 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. Detailed descriptions of the optimization functions are shown in TABLE 5-1.

{| class="wikitable" style="margin:auto; background-color:#eeeeee"
|+ TABLE 5-1 List of Used Zemax Opticstudio Optimization Functions
|-
!  style="background-color:#8d3c3c; color: #ffffff"| Type !! style="background-color:#8d3c3c; color: #ffffff""| Zemax Description !!style="background-color:#8d3c3c; color: #ffffff""|  Location!!style="background-color:#8d3c3c; color: #ffffff""|  Application
|-
| REAY|| Local real ray y-coordinate in lens units at the surface defined by Surf at the wavelength defined by Wave. See “Hx, Hy, Px, and Py”. || Surface 6: collimating lens || Target beam diameter at the collimating lens.
|-
| FTGT|| Full thickness greater than. This boundary operand constrains the full thickness of surface Surf to be greater than the specified target value. The full thickness is computed at 200 points between the vertex and edge along the +y radial direction, including the sag of the surface and the sag of the next surface. This operand has a mode flag. Mode = 0 (default) to use Mech Semi-Dia and Mode = 1 to use Clear Semi-Dia value displayed on the Lens Data Editor. The operand is useful for constraining surfaces which do not have their minimum or maximum thickness at the center or edge, but at some intermediate zone. See FTLT || Surface 3: expanding lens || Constrain minimum distance from the expander.
|-
| FTLT|| Full thickness less than. See FTGT|| Surface 3: expanding lens || Constrain maximum distance from the expander.
|-
| EFLX||Effective focal length in the local x plane of the range of surfaces defined by Surf1 and Surf2 at the primary wavelength || Surface 1-6, 2-3, 4-6 || Use as monitor for lens and who system focal length.
|}



Considering the paraxial lenses are only simplified models of real lenses, in which the thickness and aberrations are ignored when calculating the beam path, there can be some errors if we apply the idealized system to the real optic design. Therefore, another model designed and optimized using a real lens is built and used to compare with the paraxial system model. Similarly to the paraxial lens system design, he beam expander design parameters are shown in Fig. 5-6, and the optimization parameters are shown in Fig. 5-5b.

[[File:Figure 4-5a.png|500px|thumb|centre| Fig 5-5a: Paraxial Lens Optimization Table]]
[[File:Figure 4-5b.png|500px|thumb|centre| Fig 5-5b: Real Lens Optimization Table]]
[[File:Figure 4-6.png|500px|thumb|centre|Fig. 5-6: Real Lens System Design]]

Comparing the paraxial lens system in Fig. 5-5a and the real lens system in Fig. 5-5b, we can find the system focal lengths are <math> 1.784E5mm </math> and <math> 347.036mm </math> respectively, which means that the collimation of the real lens system is much worse than that of the paraxial lens system. In order to better visualize the difference, we put another paraxial tooling lens with a 100mm focal length at the image surface of each system. As shown in Fig. 5-7a and Fig. 5-7b, the real lens system has much more aberration.

[[File:Figure 4-7a.png|500px|thumb|centre| Fig 5-7a: Collimation Test: Paraxial Lens]]
[[File:Figure 4-7b.png|500px|thumb|centre| Fig 5-7b: Collimation Test: Real Lens]]

In addition, Zemax provides other tools to analyze aberrations of optic systems. The Ray Fan, Optic Path Differences (OPD), and Spot Diagrams are plotted in Fig. 5-8. Comparing the plots between the paraxial lens system and the real lens system, we can see the higher order curves for the real lens Ray Fan and OPD in both collimated (Fig. 5-8a and 5-8b) and focused (Fig. 5-8c and 5-8d) systems, meaning the spherical aberration is affecting the wave front. We can also see the clear polarized Spot Diagram pattern in Fig. 5-8b which is also caused by the spherical aberration.

[[File:Figure 4-8a.png|600px|thumb|centre| Fig 5-8a: Aberration Analysis: Collimated Paraxial Lens]]
[[File:Figure 4-8b.png|600px|thumb|centre| Fig 5-8b: Aberration Analysis: Collimated Real Lens]]
[[File:Figure 4-8c.png|600px|thumb|centre| Fig 5-8c: Aberration Analysis: Focused Paraxial Lens]]
[[File:Figure 4-8d.png|600px|thumb|centre| Fig 5-8d: Aberration Analysis: Focused Real Lens]]

=== 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.

[[File:LightTool Setup 1.png|350px|thumb|centre|Fig. 6-1: Setup of a simplified UV-LAI system in LightTools.]]

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, <math>n_2/n_1</math>, defined as 0.98. The optical density, <math>D</math>, 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 <math>n_2/n_1</math>=1), representing the ground truth level of illuminance as discussed in the background section.

[[File:sample lighttools.png|500px|thumb|centre|Fig. 6-2: Sample illuminance response from Monte Carlo simulations using the simplified UV-LAI setup in LightTools. Note that the illuminance is normalized and shown in false-colored.]]  

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 (<math>n_2/n_1</math> 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 <math>n_2/n_1</math>. 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 <math>n_2/n_1</math> = 1 to 0.99, suggesting that even a small change can have strong contribution to errors during the interpretation of absorbance.  

[[File:para1.png|550px|thumb|center|Fig. 6-3: Normalized illuminance results of the absorption image at varying levels of <math>n_2/n_1</math>.]]

[[File:para1_data.png|550px|thumb|center|Fig. 6-4: (Left): absorbance response along y = 0 mm computed from the simulations at varying levels of <math>n_2/n_1</math>. (Right): relative error at different <math>n_2/n_1</math>. Note that the x-axis is normalized by the radius of the sphere, which is 1 inch.]]

To explore the effect of ray divergence angle (<math>\theta</math>) on the interpretation of absorbance distributions, a 3<math>^\circ</math> divergence was introduced in the light source in LightTools, and the resulting absorbance distributions of the ground truth, <math>\theta</math> = 0, and <math>\theta</math> = 3<math>^\circ</math> 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.

[[File:para2.png|350px|thumb|center|Fig. 6-5: (Top): absorption images of the <math>\theta</math> = 0 and <math>\theta</math> = 3<math>^\circ</math> cases. (Bottom): absorbance response along y = 0 mm computed from the simulations at varying levels of <math>n_2/n_1</math>. Note that <math>n2/n_1</math> = 0.98 and <math>D</math> = 0.02/mm was used in the two sets of simulations, and the x-axis is normalized by the radius of the sphere, which is 1 inch.]]

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.

[[File:correction lens.png|550px|thumb|center|Fig. 6-6: CODE V schematic of the correction lens design.]]

[[File:CL_lighttools.png|350px|thumb|center|Fig. 6-7: LightTools setup of the correction lens design.]]

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.

[[File:CL_background.png|450px|thumb|center|Fig. 6-8: Background images from cases with and without correction lens and the ideal uniform background reference. <math>n2/n_1</math> = 0.98 was used in the simulations with refraction.]]

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.

[[File:CL_data.png|550px|thumb|center|Fig. 6-9: (Left): absorbance profiles of the simple spherical test gas sampled along y = 0 mm obtained from simulations with and without the correction lens in LightTools. The x-axis is normalized by the 1-inch radius of the spherical test gas. (Right): absorbance profiles of the 2-sphere test gas geometry sampled along y = 0 mm obtained from simulations with and without the correction lens. The optical density of all refracting cases are 0.02/mm.]]


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.

[[File:all_setup.png|350px|thumb|center|Fig. 6-10: LightTools setup of a UV-LAI system with a real beam expander and a correction lens.]]

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. 

[[File:ALL_IMGS.png|500px|thumb|center|Fig. 6-11: (Top): background and absorption images of a UV-LAI system with a perfectly collimated light source. (Bottom): background and absorption images of a UV-LAI system with a real beam expander. <math>n_2/n_1</math> = 0.98 and <math>D</math> = 0.02/mm were used for all simulations with refraction through the test gas.]]

[[File:ALL_DATA.png|500px|thumb|center|Fig. 6-12: Absorbance profiles sampled along y = 0 mm (across the centers of the 2 spheres) of the UV-LAI system with a real beam expander (left) and a perfectly collimated light source (right).]]

== 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 <math>n_2/n_1</math> 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 3<math>^\circ</math> 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.

== Reference ==

[1] Brummund, U., Cessou, A., and Vogel, A., “PLIF imaging measurements of a co-axial rocket injector spray at elevated pressure,” Symposium (International) on Combustion, Vol. 26, Elsevier, 1996, pp. 1687–1695.

[2] Sadanandan, R., Stöhr, M., and Meier, W., “Simultaneous OH-PLIF and PIV measurements in a gas turbine model combustor,” Applied Physics B, Vol. 90, No. 3-4, 2008, pp. 609–618.

[3] Combs, C. S., Clemens, N. T., and Danehy, P. M., “Visualization of capsule reentry vehicle heat shield ablation using naphthalene PLIF,” International Symposium on Applications of Laser Technologies to Fluid Mechanics, 2014.

[4] Niu, Z., Qi, H., Zhu, Z., Ren, Y., He, M., and Gao, B., “Nonlinear multispectral tomographic absorption deflection spectroscopy based on Bayesian estimation for spatially resolved multiparameter measurement in methane flame exhaust,” Fuel, Vol. 357, 2024, p. 129981.

[5] Hanson, R. K., Spearrin, R. M., and Goldenstein, C. S., Spectroscopy and optical diagnostics for gases, Vol. 1, Springer, 2016.

[6] Krish, A., Streicher, J. W., and Hanson, R. K., “Spectrally-resolved ultraviolet absorption measurements of shock-heated NO from 2000 K to 6000 K for the development of a two-color rotational temperature diagnostic,” Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 280, 2022, p. 108073.

[7] G. P. Lachance, S. D. Niyonambaza, É. Boisselier, M. Boukadoum and A. Miled, "Critical Comparison of Sequential and Non-sequential Modes Usage for the Design of a Czerny-Turner Spectrometer," 2020 18th IEEE International New Circuits and Systems Conference (NEWCAS), Montreal, QC, Canada, 2020, pp. 162-165, doi: 10.1109/NEWCAS49341.2020.9159788.