Proposal for Integrating a "Generalized" VLS Grating Calculator" ( Jiatai Feng)

[srio]

Allow me to briefly summarize the key points. The ray-tracing method provides a direct approach to calculating groove parameters without requiring mathematical derivation of the light path function under Fermat's principle. Secondly, coma-free focal positions can be efficiently determined for different photon energies using Powell's hybrid method (sub-second per energy point, utilizing the fsolve function in MATLAB/Python). Additionally, the ray-tracing method offers novel insights for determining the acceptance angle limit and optimal grating length.

Therefore, I would like to share this method/code with SHADOWOUI users and potentially contribute to OASYS development. However, I am unsure how to precisely extract the intersections between rays and mirror surfaces, as well as the associated incident angles, from OASYS. These are needed to calculate the diffraction angle (β) once the designed focal position is determined.

Could you assist me in integrating this VLS-G calculator into the OASYS platform?

Thank you for your time, and I apologize for any inconvenience.

Best regards,

Jiatai Feng

[srio]

Dear Jiatai Feng,

Thanks for your email and your proposition to contribute to shadowoui/shadow4.

It looks very interesting your approach.

I would suggest:

1. use shadow4 and not shadowoui (shadow3). All new developments are done in shadow4

2. It would be nice that you share a script with an example, and then we can discuss a possible integration in oasys-shadow4

3. Use the github issues for communicating (I opened an issue and copied this, please answer there: Proposal for Integrating a "Generalized" VLS Grating Calculator" ( Jiatai Feng) #54 )

Concerning your questions:

You can extract the info from the "footprint" beam. For example, in a oasys script:

import numpy
beam = in_object_1.footprint
print(dir(beam))
print("Intersection points: \n", beam.rays[:,0:3])
print("Diffraction angles [deg, to normal]: \n", numpy.degrees(numpy.arccos(beam.rays[:,5])))

Note that the calculation of the diffraction angle is based in the fact that the grating is plane, so the normal is always (0,0,1). This is not the case in curved surfaces. In this case, the normal is not the z axis. For the moment the normal it is not stored, I need to implement the way to extract it.

Best regards,

Manuel

[DavidFeng-RIXS]

Dear Prof. Manuel,

Thank you for your kind reply and support, sincerely.

Attached is an example script written in a Jupyter Notebook using Python. In this ZIP file, I have also included an example of a varied-line-spacing (VLS) cylindrical grating calculation based on the conventional light path function under Fermat’s principle. This should serve as a useful reference for verifying the code. To run the code, you can execute the LPF (Light Path Function) first, which will generate a data file (*.xlsx) for comparing energy-dependent focal positions derived from the ray-tracing method.

Regarding the ray-tracing method, please note that I am currently working on integrating diffraction/reflection efficiency calculations and the optical layout design into a unified environment for training an AI agent. Since the efficiency is written in Matlab, the following code is mainly developed with Matlab. Therefore, the Python code provided is not the most up-to-date version (I attempted to update it but encountered some issues). This is also why it took me some time to reply—I wanted to add annotations to make the code easier to understand. I apologize for any inconvenience this may cause.

In practice, using vector notation simplifies and streamlines all calculations in my opinion. Specifically, the incident angle (α) between the incoming ray and the surface normal, as well as the diffraction angle (β) between the diffracted ray and the normal, are elegantly handled. The accurate description of the normal (or tangent) direction is critical for VLS grating calculations based on the ray-tracing method. I tried some time using OASYS but I dont know it works. I will try it again.

To determine the normal direction, I used the following two approaches:

For a curved surface with a mathematical description (i.e., y=f(x)), the normal direction can be calculated from the partial derivative with respect to x ((k_tang = dy/dx = df(x)/dx, k_norm = -1/k_tang ). This method is used in the early version of the Python code.

To improve generalizability, I introduced finite element method (FEM). Here, the normal direction can be easily calculated if a 2D mesh grid is provided. However, this approach has a significant drawback: the computational time for finding ray-surface intersections becomes extremely long. While local approximation and zooming can speed up the process, it remains inefficient. ;-( Nevertheless, this method might offer new insights into defining the minimum threshold for spatial resolution in mirror surface diagnostics and manufacturing accuracy.

Thank you again for your interest and help, sincerely.

Best regards,
Jiatai Feng

Ray_tracing and_LPF_VLSSG_example.zip

[srio]

Helo Jiatai,

I had a look at your system and ideas.

I first checked the computation of the VLS parameters. I checked the case of flat grating (setting a big big R). The coefficients are the same as we obtain using the Oasys VLS preprocessor. Good! I created a workspace with your grating. A first branch is the plane grating.

Question: Why you want to use a cylindrical grating and not a plane grating? I simulated the cylindrical grating with your VLS coefficients and it works, but the results are identical with the plane grating with the appropriated coefficients (see workspace).

I understand your interest in combining ray tracing with efficiency calculations. We do not have efficiency calculations in shadow4. They are complicated and time consuming calculations and need parameters not available in shadow4 (like the ruling profile). What we could do (not yet implemented but easy to do) is to load a lookup table with the map of efficiency vs angle and energy. You could calculate that table with external programs. In many typical cases the efficiency is quite constant so typically we correct the final flux by an efficiency coefficient (typically 0.2-0.8).

Concerning ray tracing, I see you want to do everything from scratch. However you can use shadow4 for that. We implemented all the equations in vector form, and we do calculate the intersections and also the normal for all rays. And they work well. You can implement your grating in Oasys and extract the data you need. With shadow4/Oasys you could create the script that could be adapted to create your AI learning sets.

Hope this helps!

06_FVLSPGM1000eVc2M2WithWithout_FENG.zip

[DavidFeng-RIXS]

Dear Prof. Manuel,

Thank you for your kind reply. I tested your OASYS script and was pleased to see that the results match those of the published "VLS-PGM Coefficient Calculator." I spent some time attempting to write a general OASYS Python script for a VLS-PGM coefficient calculator in an optical layout comprising an upstream cylindrical mirror and a VLS-PGM (where the incident angle of the cylindrical focusing mirror is equal to that of the VLS-PGM, avoiding complications from mirror rotation that would alter the incident angle). Unfortunately, I failed because I couldn’t extract the incident angle at the VLS-PGM. Finding out my mistakes, it will require more time.

Regarding your question: "Why use a cylindrical grating instead of a plane grating?"
In your OASYS script, since the optical system’s magnification (r₁/r₂ and Cff) in the example remains the same, the identical results serve as excellent validation of the ray-tracing-based VLS grating approach.

Beyond the fact that several successful soft X-ray RIXS spectrometers (e.g., ID32 at ESRF, I21 at Diamond) use cylindrical VLS gratings, there are plaussible additional reasons for preferring cylindrical VLS gratings instead of plane gratings (in my opinion):

1. Zero-Order Focusing: Cylindrical gratings provide zero-order focusing, enabling simultaneous SASE beam transfer (for sample probing) and photon energy spectrum diagnostics (I₀ spectrum). My main goal is to develop a cavity-based soft/tender X-ray FEL beamline with pink-light (or SASE mode) adapted spectrometers for single-shot spectrum imaging (e.g., 2D-RIXS with spatial and energy resolution). By measuring the SASE light source’s I₀ spectrum, we may deconvolve the mixed RIXS spectrum (where the energy-spatially resolved spectrum measured with multi-photon-energies is the sum of the RIXS spectra from the different photon energies) and develop ghost imaging techniques for multi-photon-energies. Such transient spectroscopy could be invaluable for studying ultrafast dynamics (e.g., attosecond/femtosecond regimes).

2. Mathematical Flexibility: Cylindrical gratings generalize applications, as a series of cylinders with varying radii can approximate nearly any mirror surface profile.

3. AI-Optimization Potential: The universal adaptability of cylindrical geometry is plaussible to realize AI-based optical design tools (e.g., an 'AlphaGo for optics') without human knowledge., Nature [2017, DOI:10.1038/nature24270]).

Please let met note that, designing a novel optical layout with VLS gratings, it requires deriving the light-path function. VLS grating parameter calculations rely on Taylor-expanded light-path-function derivations under Fermat’s principle. The key challenge lies in treating the projection between the upstream cylinder and the VLS grating (as in the Hettrick-Underwood design). Ray-tracing simplifies this mathematical process, because the light propagation respects with the Fermat’s principle. Therefore, the use of the ray-tracing will facilitate to do novel optical design.

For multi-optical-element systems, planar VLS grating suffices, such as in the Hettrick-Underwood design used in SEXTANTS at SOLEIL and 2-ID at NSLS-II. However, this kind of design (namely, a upstream cylindrical focusing mirror and a VLS plane grating) have a huge difference in coma correction terms compared to SX-700-type monochromators.

Best regards,
Jiatai Feng