Hi @Liu,
Your matrix approach for handling grating imperfections is clever, but I think there's a more efficient way to tackle this.
Before diving into custom code, check out the Optometrika toolbox - it handles the basic ray-surface interactions and lens propagation, so you can focus on the grating physics. There's also GD-Calc for rigorous diffraction calculations if you need that level of accuracy.
You can find Optometrika here: https://www.mathworks.com/matlabcentral/fileexchange/45355-optometrika
And GD-Calc here: https://www.mathworks.com/products/connections/product_detail/gd-calc.html
Better Surface Representation
Instead of storing two complete matrices and subtracting them, try this approach: Start with your ideal grating surface as a mathematical function. Then add perturbations directly. For your "slightly curved and unevenly spaced" rulings, model them as:
- Spacing variations: local period changes across the grating surface
- Ruling curvature: lateral deviations from straight lines (use polynomial or spline functions)
- Surface roughness: random height variations
Combine these into a single surface representation. This is much more memory efficient and easier to work with than separate ideal and real matrices.
Reflection vs Diffraction Clarification
When you mention the beam "reflects" from the grating, I assume you're dealing with a reflection grating operating in diffraction mode (not simple specular reflection). The physics is similar to transmission gratings but with different geometry. Each ray will diffract into multiple orders, though you'll probably focus on the primary diffraction order for your analysis.
The Ray-Grating Physics
Here's the key part you're asking about. For each ray hitting the grating: 1. Find where it intersects your perturbed surface 2. Calculate the local surface normal at that point (critical for curved rulings) 3. Apply the grating equation using the local period at that specific location 4. For reflection gratings, the angle relationships are: sin(theta_diffracted) - sin(theta_incident) = m * wavelength / d_local
The critical insight is that d_local (the local grating period) varies across your surface due to manufacturing errors. Curved rulings also change the local surface normal, affecting the diffraction angles.
Ray Trace Setup
Your overall flow should be: Parallel beam → grating interaction → lens → screen convergence
For the parallel beam, space your rays according to your sampling needs - I usually start with 5-10 rays per grating period, but increase this where you have larger surface errors.
For curved rulings specifically, you'll need higher ray sampling because the local surface normal changes more rapidly. The curvature creates additional aberrations beyond simple spacing errors.
After grating diffraction, use standard ray transfer matrices for the lens. The thin lens matrix works fine for most applications: https://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis
Modeling Your Specific Imperfections
For "slightly curved rulings": * Define each ruling as a smooth curve (polynomial or spline) rather than a straight line * This affects both the intersection calculation and the local surface normal * Typical curvature might be parabolic with small deviations from straight
For "unevenly spaced rulings":
- Make the period a function of position across the grating
- Could be systematic (linear chirp) or random variations
- Keep changes smooth to avoid unrealistic scattering
Practical Implementation Tips
Start simple - do everything in 2D first (just the x-z plane). Once that's working, extending to 3D is straightforward. Use adaptive ray sampling - more rays where surface errors are largest. Compute this from the gradient of your surface error function. For performance with large gratings, consider dividing the surface into patches and using different ray densities for each patch based on local surface complexity.
What You Should See
If everything's working correctly, you'll observe: * Focal spot broadening from wavefront distortion * Asymmetric spot shapes from systematic errors * Reduced peak intensity compared to perfect grating * Possible ghost images if you have periodic spacing errors * Stray light distribution from random surface variations
Screen Analysis
To quantify your results: * Measure focal spot diameter (full width at half maximum) * Calculate Strehl ratio (peak intensity compared to perfect case) * Analyze wavefront quality using Zernike polynomial fits if needed * Plot point spread function and encircled energy curves
Computational Considerations
For large gratings with many surface points, this gets computationally intensive. Consider: * Parallel processing for independent ray calculations * Efficient surface interpolation methods for smooth error functions * Memory management for large ray arrays
Validation Strategy
Always test with a perfect grating first - should give you a perfect focus at the design wavelength and angle.
Then add one type of error at a time: 1. Pure spacing errors (keep rulings straight) 2. Pure curvature (keep spacing perfect) 3. Combined effects 4. Random vs systematic errors
This lets you understand which imperfections cause which optical effects.
Quick Start Approach
I'd recommend: 1. Grab the Optometrika toolbox for basic ray propagation infrastructure 2. Start with 2D simulation and simple spacing errors only 3. Add ruling curvature once the basic physics is working 4. Extend to 3D and add more complex error patterns 5. Validate each step against known optical principles
This type of simulation can be tricky to get right, but once it's working, it's incredibly useful for understanding real grating performance and tolerancing manufacturing specifications.
Feel free to reach out if you hit any specific roadblocks during implementation. The ray-surface intersection calculations with curved rulings can be particularly challenging.
References:
- Optometrika Toolbox: https://www.mathworks.com/matlabcentral/fileexchange/45355-optometrika
- Ray Transfer Matrices: https://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis
