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Results for high frequency approximations (Wave propagation)

  • 12 September 2021
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Hello,

I am a new user of the OpticStudio software which arouses my curiosity!

The wave propagation can be solved with several models and the software offers several possibilities. The model based on the astigmatic Gaussian beams (beamlets) is very well interesting.

But, I found few years ago the relevant paper:

Gosse L., James F., Convergence results for an inhomogeneous system arising in various high frequency approximations, Numer. Math. 90: 721–753 (2002).

It is enough hard because the mathematics formulations are theoretic. However, the issue solving gives several possibilities to find the phase and the amplitude of the light wave together.

Numerical values for the phase and the amplitude of the smooth wedge.

Therefore, do we have this possibility with the OpticStudio? Compute the phase in order to see transparency images for instance?

Thank you in advance for your answer.

Benoît.

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Best answer by Sandrine Auriol 27 September 2021, 11:30

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Dear Sandrine and Michael,

It seems to the adding reference on the WebSite of the paper is not present. Its link is: https://www.sciencedirect.com/science/article/abs/pii/S1572100021000181?via%3Dihub.

Kind regards, Benoît.

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Hi Benoit!

This is brilliant. Thank you.

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Hi Benoît,

That’s a great news and congratulation to the publish in Scientific Reports!!

And thank you for adding the information in Acknowledgements. I’m happy Zemax can help a little here.

I’m quite overwhelmed by many tasks now, but I really want to take time to read it. Congrats for the great work again and thank you for sharing it to us!

Best regards,

Michael

Dear Michael,


The paper concerning our previous exchanges is published now in the “Scientific Reports” (https://www.nature.com/articles/s41598-022-24176-8). The authors thank the ZEMAX developer team for their help and especially to better know the OpticStudio Software (Thanks are in the paper).


But the adventure not stops, you will find a forum on WebSite of the Springer Nature to exchange if you wish. I will drop on the WebSite a new interesting reference to begin the exchanges.


Best regards,
Benoît.

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Hi Benoît,

Congratulation for the progress! This looks cool. Am I correctly understanding these are the irradiance of a collimated input beam at each “slice” as you mentioned before? The reason you say they are fluorescence image is because the excited light should be proportional to the irradiance, correct?

This looks interesting to me. It’s a 3D intensity distribution I(x,y,z) if I don’t mistook. We could convert it to a light source and simulate it in ray-tracing way with a real or ideal microscope model to know how it would look like in a microscope system. Just brainstorm for fun.

This looks nice and congratulation again!

Best regards,

Michael

Hello Michael,

We advanced with the reconstruction of the biological tissue by using the Gosse’s algorithm.

The reconstruction of a image stack is possible lighting by a parallel ray beam. These images contain small structures. You have two 3D views (OpenGL), bellow:

First View.

 

Second view.

 

Best regards, Benoît.

PS. They are fluorescence images.

 

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Thank you, Benoît.

That’s interesting and I will try to have a read. I also forwarded this to one of my colleagues who is still a PhD student doing research related to optical neural network.

Yes, I agree this is related to diffractive optics. Normally I’m confident when I want to build the macro system for the biomedical optics related application, but it’s a little tricky when it comes to exactly modeling the light interaction with those micro structures, like nuclei. The book is very interesting. It’s a good reference to me if I want to understand the requirement of this kind of systems more, and thank you again for sharing the guidance.

 

Best regards, Michael

Great, Michael!

I know not very good as you the diffractive optics. However, the “Nature photonics” journal edited his “Volume 15 Issue 5” with a cover showing “Diffractive optical computing” (Nature Photonics volume 15, pages 367–373 (2021)).


“Artistic impression of optical computing performed by modulating the incident light with layers of diffractive structures, comprised of programmable liquid crystal array. A photodetector array then converts diffracted photons into electrons to realize a reconfigurable optoelectronic processor.”
 

I not read this paper but it must be interesting, the diffractive optics is surely future technology. I will look a bit this.

Our current study is nuclei inside the biological tissues which may act as micro-lenses (DD. Nolte, Optical interferometry for Biology and Medicine, 2012, Part of the Bioanalysis book series, Springer). I think this issue should be also diffractive optics as same way.

Warm regards, Benoît.

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Thank you, Benoît!

I have forwarded this to development team for their information. Although I don’t think there will be an action any soon, but I think we should keep an eye on this kind of potential theories too. As the manufacturing method improves, diffractive optics or small structure would gradually become  commercially possible. And this kind of algorithm would be important to simulation in the future. I’m actually thinking this might be useful to the so-called diffractive lens simulation. But I don’t have enough knowledge about how it works, so I probably should not assume too much before I read it carefully.

And yes, that would be wonderful if you can share update when you have progresses to the paper or any publish!

Best regards, Michael

Hi Michael,

Thank you for your returns.

I not know the benefice for the industrial partners but it would be great if the OpticStudio could introduce the Gosse’s algorithm.

Agreed, it is well the Zemax citation. I will see with the co-authors to ask the shared mode with the journal. In this way, I think the exchanges could be enlarged.

I stay ready to carry on our discussion, if you wish.

Best regards, Benoît.

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Hi Benoît,

Thank you for the sharing and discussion again.

POP hasn’t updated its fundamental algorithm for recent years but this makes me feel it might time to move forward. I will share this discussion to development team in case it’s useful to them.

For the reference, I have checked internally and it looks like we've never given specific instructions for citing Zemax in published papers. People will often reference the OpticStudio software by Zemax LLC as a tool used to produce some of the images in a paper. I have also seen some papers discussed the difference of its algorithm to that in OpticStudio in introduction.

I have also got some suggestions, but please consider these only if you are interested. The suggestions are that maybe you could post a summary about the paper on the community when it’s available, and host a “ask me anything” on thread for that. Otherwise, I think this discussion is already great. I’ve learnt a lot and believe it’s so to other readers too. Thank you!

Best regards,

Michael

Hi Michael,

1. Yes, you are right. I not known this paper because I purchased the Pierre Pellat-Finet book (In French, unfortunately!). Strangely, the fractional Fourier transform is used also for the corpuscular physics. The relationships between the optical and corpuscular physics are closed. In the Gosse’s paper, you can remark that the author applies also his model for Schrödinger’s and Helmhotz’s equation, the wave physics stays very logical! It seems to me the programming with spherical surfaces as POP must be better for the simulation results.

2. Thank you for this detail of the POP programming. The Rayleigh length is the theory limit between the small and the large changes of the Gaussian beam width. The switch between the two models seems to me interesting relative to this limit.

3. The Lax-Friedrichs scheme is a finite difference time domain algorithm (FDTD). My programming is better now, but its is not very sure for all object types. The tool stays just a prototype to appreciate the power of the optics equations to solve simultaneous the phase and the irradiance and apply it to biological issues (see the image bellow).

One Gosse’s image extracted of the bubble cluster.

In common use, it seems to me the principle of scalar field is correct when all partial derivative equations governing the electromagnetic field has the same expression. Indeed in this case, you can choose a field component (electric or magnetic) as the scalar field. Luckily, the asymptotic equations of the optics keep this vector property and we can extend the Gosse's algorithm to the vector format for instance. It would be surely not easy but it seems possible!

4. I will propose a letter in a journal concerning the Gosse’s algorithm, it is not obvious that this paper will be accepted. However, I wish to thank the Zemax team to ours fruitful exchanges in this paper. Do you have a particular formulation for the Zemax company?

Kind regards, Benoît.

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Hi Benoit,

I’m very sorry for late reply.

  1. Thank you for sharing again. That is a lot of information to me. I think what you mentioned is that introduced in this paper. I had read a little but obviously I need more time to understand it. The change of plane based field to spherical based is interesting. In POP, we also reference the phase to a sphere in order to decrease required grid sampling rate, but I think they are totally different technique.
  2. I would say the Angular Spectrum propagator is mainly good in any case where the beam size doesn’t change too much before and after the propagation. At least it is so in POP’s implementation. For a Gaussian beam, this means it’s only applicable for the beam propagation inside of the Rayleigh range. When the beam start or end at outside of the Rayleigh range, POP will switch to use Fresnel propagator.
  3. The use case you showed is definitely something that POP cannot handle. POP mainly considers the case of beams not too diverging and the lens curvature not too large. Some small particles (water bubble) are something POP’s algorithm cannot handle for now unfortunately. It’s amazing you can you combine these tools and make the prototype. I hope one day we could have this kind of feature to support your case. If you didn’t show me Goose’s method, I would say this kind of structure might require FDTD or BPM, but I might be wrong. One thing I noticed is that it looks like Goose’s method is based on scalar diffraction.  If the water bubble is too small, I wonder if it becomes important to consider vector diffraction.

Thank you for the information and sharing your works again. I’m sorry that I feel POP may not be very useful in some part of your system, but please let me know if you have any questions to the method we used in POP!

 

Best regards,

Michael

Hi Michael and Sandrine,

Thank you very much for your interesting comments.

1. The ImageJ plugin exploits the Fresnel’s propagation only from images. I think OpticStudio is able to simulate this propagation through optical device, it is a main difference. But I yet must work to adapt exercises for the students. Another analysis is the “metaxial approximation” theory by Georges Bonnet and Pierre Pellat-Finet in order to move away a bit of the “paraxial approximation” (based on the signal processing).

2. It seems to me the angular spectrum algorithm is based on the Fourier basis and uses when the distance with the source is small (the mode “automatic” of the ImageJ plugin takes account this distance in order to switch between "angular spectrum" and "Fresnel propagation").

3. Even if I know the optical equations since a few years ago, the Gosse’s algorithm is also new for me. I programmed the base of this algorithm in C language and I use “Octave software” (version freeware of Matlab) and a bit ImageJ for the illustrations. This prototype is not yet very robust because I not taken fully into account the numerical divergences. It is exactly the Gosse’s work in his paper. Nevertheless, this way seems to me promising and I hope write a simple letter in a journal to show the power of this algorithm. Unfortunately, this paper type (between practice and theory) is not very easy to publish.

Our issue is to understand the path of light through samples (see below, please)

Three water bubbles in the space whose the centers are arranged in an equilateral triangle.

Three water bubbles in the space whose the centres are arranged in an equilateral triangle.

It is the last simulation of a image stack by means of the Gosse’s prototype.

Warm regards, Benoît.

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Hi Benoît,

Thank you for sharing the information!

  1. The plugin is very interesting. Although it still depends on the implementation, but I agree then this plugin is at same accuracy to POP in theoretical level, at least for simple propagation.
  2. I didn’t know into this details and thank you for sharing the theory behind the POV rays! I think you are correct that Fresnel has some paraxial approximation. Angular spectrum in theory doesn’t have paraxial approximation, but it’s only numerically applicable if the beam is not too diverging.
    1. Note all these are limited to scalar diffraction assumption. It’s correct if we only propagate the beam in free space (homogeneous and isotropic). When the beam passes through a boundary of AIR/Glass like with a lens, the probing rays are used to transfer the beam from “this side” to the “other side”. To this point, the Goose’s method looks interesting and thank you for sharing again.
  3. I don’t know Goose’s method very well, but it looks to me it’s a strength if it can handle for highly curved surface very well, although I wonder if there is any drawback on the other hand.

Thank you!

Michael

Hello Sandrine,

Just a small demo in images.

When you run imageJ with the “numerical plugin” (version 1 or 2), you can run this plugin as below:

ImageJ with the plugin

From a image of the equilateral triangle:

Source image

You obtain:

Result image after the propagation (1m)

the irradiance image with the colormap named “red hot”.

Thank you Sandrine,

Benoît.

PS. I attached a ZIP file with the images.

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Thank you Benoit. I didn’t know about that plugin. I have copied the link here if anyone is interested: https://unal-optodigital.github.io/NumericalPropagation/.

Hi Michael and Sandrine,

Thank you very much for yours answers.

1. The “numerical propagation” plugin of the imageJ software is able to use the “angular spectrum propagation” and/or “Fresnel propagation”.

Numerical propagation module

Ours students work with this plugin to simulate the Fresnel’s holograms, for instance:

Hologram of the “R” char

I not used yet OpticStudio for the holograms, I am a Zemax beginner! Surely, OpticStudio is better and I will study your tool to propose new exercises for the students. Your OpticStudio tool is very well!

2. I agree for you. I not know perfectly your tool but I think it are two independent modules.

The POV module is surely based on the common Descartes’ law (see below in Frenet’s dynamic coordinate system)

Common Descates’ law

Therefore, only the light rays taken account.

The POP module is based on the wave propagation (angular spectrum, Fresnel) but I think you must also take account the paraxial assumption (required for FFT) . For example for the Fresnel formula, the convolution formula must be used (see below with scalar electric field “psi”).

Fresnel’s convolution formula (linear systems)

3. Thank you for your experimentation, it is interesting. I wish also to test OpticStudio for my research work concerning the histological imaging. The goal is to understand how the image training by the biological specimen. Perhaps, the Gosse's paper could give a answer because it seems to me it is possible to compute a image from several intensity plans inside of specimens. The advantage of this method is that the computations of the phase and the irradiance perform together without the paraxial assumption.

Have a nice day, Benoît.

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Hi Benoit,

Please allow me to reply as I have also checked this discussion.

I have a few comments to what have been discussed so far.

  1. About the method used in Zemax OpticStudio, it’s mainly a mix of the so-called Fresnel propagation and the Angular spectrum propagation. The method in ImageJ based on FFT can be also considered as Fraunhofer propagation. Fraunhofer propagation is the simplified version of the Fresnel propagation. Fraunhofer propagation only works when the beam is propagated to far field or when it’s focused by a lens. I think the Physical Optics Propagation (POP) in Zemax OpticStudio should be more accurate when it’s far from the above cases. If you are interested in knowing more details of Fresnel and Fraunhofer diffractions, you can refer to the book Introduction to Fourier Optics by Goodman.
  2. As far as I know, POV rays is based on pure geometric method. It doesn’t provide phase data and even the coherent intensity data. I think it’s hard to compare its result with POP.
  3. About the example file you provided in spheric_dioptre.zar, I agree with Sandrine that this is too challenging for Physical Optics Propagation of ZOS. The main problem is the lens you build in the system is with too high curvature that introduces very large phase slope. Those what you called bubbles are aliasing. There are severe aliasing in the phase profile. This basically means any propagation result after this surface is not trustable. The only method to solve this issue is to increase the beam sampling. I have tested beam sampling with 16384x16384, but have no luck. Unfortunately, I think the conclusion is you cannot use POP with this highly curved lens. If you mainly want to compare the result in POP with your code, I suggest you try a milder curvature for your lens.

 

 

Please let me know if you have any more questions.

Thank you!

Hello Sandrine,

This paper is a bit hard because the authors are very good mathematicians. I unscrambled few demonstrations for the programming of their method in 2D which diverges for some cases but which expands to the 3D. I therefore retain only the first paragraphs and the figures. I remade nearly their figures (see below by centring the smooth wedge and taking "z" as optical axis).

Their optical component
The phase image of the smooth wedge
The irradiance image of the smooth wedge

Tomorrow, I will study the POPI operand in ZPL and API. I not known yet these operands. I should have surely few questions!

Your beautiful phase image seems to me correct because I find again there bubble clusters due to phase computations, thank you.

Warm regards, Benoît.

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Hi Benoit

Sorry I haven’t read the paper yet, but are the plots you are showing along the propagation distance?

If that is the case, I suppose you could try to get similar plots using the POPI operand in ZPL or API. With the API, you can  also read the results directly. You will need to save the beam at all the Z steps you wish to have.

That being said, I am not sure that the POP results are correct as we can see that the phase sampling doesn’t look great. This is the phase sampling at surface 4 for the spheric_dioptre.zmx. We can also see that we have warnings in the Prop Report of POP:

I looked at the sampling needed in your case (see the article https://support.zemax.com/hc/en-us/articles/1500005488641-Using-Physical-Optics-Propagation-POP-Part-3-Inspecting-the-beam-phases) and I uploaded the merit function in your system and it gives me a sampling of around 16,000. 

 

I haven’t tried to do the calculation yet as it is quite long.

At this point I think I’ll ask one of my colleague to check because I’m not sure the software can do what you are planning to.

Hello Sandrine,

Thank you, your experimentation agrees me very well.

Yesterday, I progressed the programming concerning the Cosse’s paper (first post).

Two simple tests:

First one is the plane dioptre (see the propagation of OpticStudio):

The first surface is the reference, the second surface is the dioptre (N-BK7 material) and third surface is the image.

The irradiance profile on the second surface computed with “Physical Optics”:

The irradiance profile on the third surface computed with “Physical Optics”:

The same test was performed with the Gosse’s method:

At the right, we show the irradiance (not amplitude) decreasing after the dioptre crossing due to the transmission coefficient (Fresnel’s coefficients).

Second one is the spherical dioptre (see the propagation of OpticStudio):

The first surface is the reference, the second surface is the dioptre (N-BK7 material) and third surface is the image.

The irradiance profile on the second surface computed with “Physical Optics”:

The irradiance profile on the third surface computed with “Physical Optics”:

The same test was performed with the Gosse’s method:

At the right, It seems to me we show the irradiance (not amplitude) with the caustics after the dioptre crossing.

Do you think it is possible to simulate the same effects (irradiance decreasing and caustics) with OpticStudio? What module should be to use?

The OpticStudio files are attached to this post.

Have a nice weekend, Warm regards, Benoît.

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I have looked at the POV ray file but I also have an error. Different one though. I’ll check if someone internally knows how to use this.

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Hello Benoit 

I have looked at the sampling of POP in the file you attached and I have adjusted it. When going through the pinhole, the irradiance looks square because it is described by a small amount of pixels so this is what I was looking for.

So I started by changing the initial array size as the beam has a very low divergence:

Then I resampled the beam after the pinhole. The file is attached.

About the POV ray, I’ll have a look.

Hi Sandrine,

Thank you for your experimentation.

Yes, I agree with you for the POP and “geometric” definitions.

The optic device is divided into two parts: 1/ planoconvex lens to build a parallel beam and 2/ a circular obstruction (radius 0.2 mm). Only, the second part is interesting for the diffraction phenomena (between surface 3 and surface 4).

Irradiance on surface 3
Irradiance on surface 3

I study more details the Gosse’s article (first post) and I began the programming to try compare the methods. It is not finished! The final goal is to simulate the diffraction of our circular obstruction. You find a phase image of a spherical dioptre lighted by a parallel beam.

Phase image (spheric dioptre)

Otherwise, I tried the OpticStudio Macro to build a Povray file. The Povray software finds a error “Problem with option setting”. I examined the generated file and It seems to me correct? (attached with this post).

Warm regards, Benoît.

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