How can I simulate PSF of incoherent light?

[snowstarlele]

Hi There,

Recently, I am working on simulating of Point spread function of high numerical aperture objective lens, according to the Richard and Wolf's mathematics representation, I can do the calculation of PSF, like transverse size or longitudinal size without any difficulty, but this formula is based on diffraction theory, not working with incoherent light. How can I calculate transverse size and longitudinal size of incoherent light PSF ?

Does anyone tell me an appreciate formula or method ?

Thanks
Best regards.

Qinggele

 

[DrDu]

You can simply average the intensity patterns of light with two orthogonal polarizations.

 

[snowstarlele]

Thanks for your reply.
As your suggestion. I've done the simulation for orthogonal polarized light. but there is still side lope, I am wondering, if the incident light is incoherent, I don't think there is side lope emerged, because there is no interference between incoming rays.
If I set random phase for each rays, in focal region, there is no PSF formed, the instead is homogeneous intensity distribution everywhere, which is not obviously wrong for light focusing. I am thinking maybe for the incoherent light focusing, the wolf's diffraction formula(based on debye approximation) no longer working.

maybe orthogonal polarized light is suitable for simulation of unpolarized light(or random polarized light).

the attachment is the PSF from orthogonal polarized light. I just add the intensity patterns of x-linear and y-linear. I also enclosed one paper which I use for simulate PSF.

Maybe I didn't catch your idea. could you explain to me in detail ?

thank you very much

 

[Andy Resnick]

Hi There,

Recently, I am working on simulating of Point spread function of high numerical aperture objective lens, according to the Richard and Wolf's mathematics representation, I can do the calculation of PSF, like transverse size or longitudinal size without any difficulty, but this formula is based on diffraction theory, not working with incoherent light. How can I calculate transverse size and longitudinal size of incoherent light PSF ?

In what sense do you mean 'incoherent'?- spatial? temporal? both?

What's the Richard and Wolf reference? I tend to use Min Gu's "Advanced Optical Imaging", with high NA lenses covered in Chapter 6.

 

[snowstarlele]

In what sense do you mean 'incoherent'?- spatial? temporal? both?

What's the Richard and Wolf reference? I tend to use Min Gu's "Advanced Optical Imaging", with high NA lenses covered in Chapter 6.

Hi Andy Resnick,

I think the both, because in my case, i just use white light source with vary narrow band pass filter. the light is focused by NA=0.6 objective.

In second post, I've enclosed one relevant paper which I use to simulate the PSF.
I think I going to read the book 'Advanced optical imaging theory'.

Regards.

 

[Andy Resnick]

I think the both, because in my case, i just use white light source with vary narrow band pass filter. the light is focused by NA=0.6 objective.

In second post, I've enclosed one relevant paper which I use to simulate the PSF.

Sorry- didn't see that attachment. I have a better understanding of what you are trying to do. I assume you are trying to understand equations (2) and (3)?

The authors gloss over a few minor details- their eq. (1) is called the 'Sine condition', and is not the only apodization condition. There are also the Herschel (h = 2f sin(θ/2)), Helmoltz (r = f tan(θ)), and Uniform projection (r = fθ), and so it is important to know which condition the lens obeys (AFAIK, Leica objectives obey the sine condition, for example).

Of more consequence is the effect of dielectric interfaces (for example, a coverslip): here, the 'high NA rays' will be grossly effected due to the different s- and p- polarization coefficients of reflectivity, resulting in significant changes to the PSF.

Both of these are handled in vectoral Debeye theory- the results are straightforward, but the expressions are rather long (Gu's book has them on 6.5.9-6.5.13); 6.5.18 is the intensity.

A few more references:

P. Török, P. Varga, Z. Laczik, and G. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation," J. Opt. Soc. Am. A 12, 325-332 (1995).

P. Török and P. Varga, "Electromagnetic diffraction of light focused through a stratified medium," Appl. Opt. 36, 2305-2312 (1997).

 

[DrDu]

Sorry, I didn't understand you correctly.
The Book by Born and Wolf may be useful. I think you could use the Van Cittert Zernike theorem:

http://en.wikipedia.org/wiki/Van_Cittert–Zernike_theorem

 

[snowstarlele]

Sorry- didn't see that attachment. I have a better understanding of what you are trying to do. I assume you are trying to understand equations (2) and (3)?

The authors gloss over a few minor details- their eq. (1) is called the 'Sine condition', and is not the only apodization condition. There are also the Herschel (h = 2f sin(θ/2)), Helmoltz (r = f tan(θ)), and Uniform projection (r = fθ), and so it is important to know which condition the lens obeys (AFAIK, Leica objectives obey the sine condition, for example).

Of more consequence is the effect of dielectric interfaces (for example, a coverslip): here, the 'high NA rays' will be grossly effected due to the different s- and p- polarization coefficients of reflectivity, resulting in significant changes to the PSF.

Both of these are handled in vectoral Debeye theory- the results are straightforward, but the expressions are rather long (Gu's book has them on 6.5.9-6.5.13); 6.5.18 is the intensity.

A few more references:

P. Török, P. Varga, Z. Laczik, and G. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation," J. Opt. Soc. Am. A 12, 325-332 (1995).

P. Török and P. Varga, "Electromagnetic diffraction of light focused through a stratified medium," Appl. Opt. 36, 2305-2312 (1997).

thanks, I also repeated this two references's results before without any difficulty, using the given mathematics formula, which based on Debey approximation. Now the problem is that for the incoherent light source, I think there is no interaction between each ray, it should be only sum of intensity in focal region. but once I used (Itotal=E1E1*+...+EiEi*+...EalphaE*alpha) instead of Itotal=(E1+...Ei+...Ealpha)(E1+...Ei+...Ealpha)*, where alpha is maximum open angle of objective lens, alpha=asin(NA/n), I couldn't get any intensity pattern, the instead is the uniform intensity distribution everywhere in focal plane, I think maybe it is because each ray point to focal region as spherical shape, so if do square for only one ray it should be get uniform amplitude everywhere(quiet personal view). I am so confused for this. do you have any experience for this ?

I do appreciate your timely help

Best regards

 

[snowstarlele]

Sorry, I didn't understand you correctly.
The Book by Born and Wolf may be useful. I think you could use the Van Cittert Zernike theorem:

http://en.wikipedia.org/wiki/Van_Cittert–Zernike_theorem

Thank you very much for your suggestion.

I going to try this method.

Regards

 

[Andy Resnick]

t<snip> I am so confused for this. do you have any experience for this ?

I had a hard time understanding this post. In cylindrical coordinates, the field E(r,ψ,z) propagating in direction 'k' at focus is:

E(r,ψ,z) = iπ/λ[(I_0+ cos(2ψ)I_2)i +sin(2ψ)I_2 j + 2icos(ψ)I_1 k,
I_0 = ∫P(θ)sinθ(1+cosθ)J_0(krsinθ)exp(-ikzcosθ)dθ
I_1 = ∫P(θ)sin^2(θ) J_1(krsinθ)exp(-ikzcosθ)dθ
I_2 = ∫P(θ)sinθ(1-cosθ)J_2(krsinθ)exp(-ikzcosθ)dθ

where i,j,k are unit vectors and there should not be confusion about the other i = √(-1) or other k = 2π/λ, J_0 etc are Bessel functions, θ runs from 0 to sin^-1(NA/n), P(θ) the apodization function, etc. In the paraxial approximation, I_1 = I_2 = 0, and the usual result is obtained.

The intensity is |E|^2:

I(r, ψ, z) = C[|I_0|^2 + 4|I_1|^2 cos^2(ψ)+|I_2|^2 +2cos(2ψ)Re(I_0 I_2*)]

Does this help?

 

[snowstarlele]

I had a hard time understanding this post. In cylindrical coordinates, the field E(r,ψ,z) propagating in direction 'k' at focus is:

E(r,ψ,z) = iπ/λ[(I_0+ cos(2ψ)I_2)i +sin(2ψ)I_2 j + 2icos(ψ)I_1 k,
I_0 = ∫P(θ)sinθ(1+cosθ)J_0(krsinθ)exp(-ikzcosθ)dθ
I_1 = ∫P(θ)sin^2(θ) J_1(krsinθ)exp(-ikzcosθ)dθ
I_2 = ∫P(θ)sinθ(1-cosθ)J_2(krsinθ)exp(-ikzcosθ)dθ

where i,j,k are unit vectors and there should not be confusion about the other i = √(-1) or other k = 2π/λ, J_0 etc are Bessel functions, θ runs from 0 to sin^-1(NA/n), P(θ) the apodization function, etc. In the paraxial approximation, I_1 = I_2 = 0, and the usual result is obtained.

The intensity is |E|^2:

I(r, ψ, z) = C[|I_0|^2 + 4|I_1|^2 cos^2(ψ)+|I_2|^2 +2cos(2ψ)Re(I_0 I_2*)]

Does this help?

Sorry for the making you so confused.

Thank for your timely replay.

I am not familiar for typing math here. so I enclosed pdf format document to here. please find the attachment.

Thanks again for your help.

Regards.

 

[Andy Resnick]

I am not familiar for typing math here. so I enclosed pdf format document to here. please find the attachment.

Thanks for the document, that does help me understand a lot better. I don't think you can simplify [4] the way you hope, and even worse I suspect that as your NA gets large, there may be numerical instabilities similar to Mie scattering results when the phase term oscillates very rapidly, if you discretize too coarsely.

Also, your [2] looks vaguely familiar- where did that come from?

Are you able to recover 'known' results if your NA is small?

 

[snowstarlele]

Thanks for the document, that does help me understand a lot better. I don't think you can simplify [4] the way you hope, and even worse I suspect that as your NA gets large, there may be numerical instabilities similar to Mie scattering results when the phase term oscillates very rapidly, if you discretize too coarsely.

Also, your [2] looks vaguely familiar- where did that come from?

Are you able to recover 'known' results if your NA is small?

Hi Andy Resnick,
Thanks for your replay.

1. for the formula [2], there is one paper from Lars Egil Helseth: "Focusing of atoms with strongly confined light potentials" explained it in detail.

2. normally, the mathematics representation based on vector debye approximation is more suitable for tight focusing where NA>0.6, for the case of extremely low NA, it does not give correct results, because polarization(vector property of light) no longer plays dominant role, the instead is using formula based on scalar theory.

do you think the mathematics representation still valid in the case of incoherent ? I don't know how can i change the formula when the incident light is incoherent.



Regards.

 

[Andy Resnick]

1. for the formula [2], there is one paper from Lars Egil Helseth: "Focusing of atoms with strongly confined light potentials" explained it in detail.

<snip>

do you think the mathematics representation still valid in the case of incoherent ? I don't know how can i change the formula when the incident light is incoherent.

If I read that paper correctly, Helseth did most of your work already: all you need are the results [8-12] and [15-18]. Since radial and tangential polarization states are orthogonal, the (incoherent) intensity is simply I = 1/2[I_r + I_t], where I_r is the intensity from radial polarization I_r = |E_r|^2 and I_t the intensity from tangential polarization. The components of E_r are given in [8-12] and E_t in [15-18].

Thanks for the reference- there's some good stuff in there.

 

[snowstarlele]

If I read that paper correctly, Helseth did most of your work already: all you need are the results [8-12] and [15-18]. Since radial and tangential polarization states are orthogonal, the (incoherent) intensity is simply I = 1/2[I_r + I_t], where I_r is the intensity from radial polarization I_r = |E_r|^2 and I_t the intensity from tangential polarization. The components of E_r are given in [8-12] and E_t in [15-18].

Thanks for the reference- there's some good stuff in there.

thanks for your replay.

If I understand clearly, your meaning is that doing simply add the two intensity patterns of radial and tangential polarization? which means that in the case of incoherent, it is equivalent to the Superposition of PSF from radial polarized beam and PSF from tangential polarized beam. yes, there is no interaction between these two orthogonal polarized lights, we can assume that it is same like incoherent. if it works, I am wandering maybe for incoherent, superposition of PSF from x-linear and y-linear polarized light also can do incoherent work.

Could you tell me further information about the assumption ? do you have any reference for that ?

thank you very much for your help.

Regards

 

[Andy Resnick]

<snip>Could you tell me further information about the assumption ? do you have any reference for that ?

I'm not sure what you mean by 'assumption'...? For example, using the Stokes vector (S0, S1, S2, S3) to describe the polarization state, the total intensity is S0. You can choose any basis states for polarization: x and y, left- and right-circular, radial and tangential, etc.

 

[DrDu]

After having proposed to average over polarizations myself first, I now rather think that you have to average over waves entering under slightly different angles or from different points (depending on the characteristics of the incoherent light source!)
Generally, I find this thread hard to follow: E.g. the article you cite does not mention point spread function.
Maybe you could just write down the expression you found for coherent light and explain it?

 

[snowstarlele]

I'm not sure what you mean by 'assumption'...? For example, using the Stokes vector (S0, S1, S2, S3) to describe the polarization state, the total intensity is S0. You can choose any basis states for polarization: x and y, left- and right-circular, radial and tangential, etc.

Hi Andy Resnick,

Thanks,

Here I've enclosed one attachment, which is according to your suggestion.
I = 1/2[I_r + I_t].

I was confused the difference between incoherent light and unpolarized light. I think I = 1/2[I_r + I_t] is more suitable for unpolarized light. do you think <<1/2[I_r + I_t]>> can reflect the incoherent but linear polarized light ? (such as the light from white light source + filter+polarizer).

thanks again

Regards

 

[DrDu]

I looked at the definition of the PSF and found that the answer to your original question is very simple: The point spread function defines the spread of light from a point source when passing through some optics. But the light from a point source is always completely spatially coherent and, if monochromatic, also temporally coherent.

 

[snowstarlele]

After having proposed to average over polarizations myself first, I now rather think that you have to average over waves entering under slightly different angles or from different points (depending on the characteristics of the incoherent light source!)
Generally, I find this thread hard to follow: E.g. the article you cite does not mention point spread function.
Maybe you could just write down the expression you found for coherent light and explain it?

Hi

Thank you very much for your post.

I tried with small angle. please find the attachment. I also put the formula for PSF.

Thanks again for your help,

 

[snowstarlele]

After having proposed to average over polarizations myself first, I now rather think that you have to average over waves entering under slightly different angles or from different points (depending on the characteristics of the incoherent light source!)
Generally, I find this thread hard to follow: E.g. the article you cite does not mention point spread function.
Maybe you could just write down the expression you found for coherent light and explain it?

Sorry forgot to put attachment. here it is.

 

[DrDu]

What are r_P, theta_P, phi_P, and theta, phi? Where is the point light source?

 

[snowstarlele]

What are r_P, theta_P, phi_P, and theta, phi? Where is the point light source?

r_P, theta_P, phi_P, image coordinate
theta is open angle of objective lens, theta=arcsin(NA/n). phi is spatial angle at lens coordinate. there is one papar from wolf explain detailly about transfrom from Cartesian coordinate to polar coordinate.

Here it is the paper . please find the attachment.

Thanks for your help.

Regards.

the meaning of formula is the supperposition of all light rays which is diffracted from lens aperture.

 

[DrDu]

Can't see the paper

 

[snowstarlele]

Can't see the paper

Does it works now? I enclosed again . It's so weird.

if it still not be able to upload, please find it on google by title "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system"

Sorry for the incontinence.

Regards.

 

[DrDu]

It didn't work but I had a look at the paper.
As I said I don't see any problem as the light from a point source is always coherent.
Incoherence enters when you want to calculate the image of an extended object or source.

 

[Andy Resnick]

<snip>do you think <<1/2[I_r + I_t]>> can reflect the incoherent but linear polarized light ? (such as the light from white light source + filter+polarizer).

If I understand you correctly, no- while you started with incoherent light, you ended up with coherent light. We may be getting off track here- are you still asking how to compute the incoherent PSF, or are you now asking something else?

 

[Andy Resnick]

<snip> But the light from a point source is always completely spatially coherent and, if monochromatic, also temporally coherent.

Usually that's correct- but polarization is another degree of freedom.

 

[DrDu]

Usually that's correct- but polarization is another degree of freedom.

Sure, but adding two intensities is not really a problem. Furthermore, the case of unpolarised light has been treated in the article by Richards and Wolf to which the OP apparently was referring all the time:

Richards, B., & Wolf, E. (1959). Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 253(1274), 358-379.

 

[Andy Resnick]

Sure, but adding two intensities is not really a problem. Furthermore, the case of unpolarised light has been treated in the article by Richards and Wolf to which the OP apparently was referring all the time:

Richards, B., & Wolf, E. (1959). Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 253(1274), 358-379.

Right- and the OP was (originally) trying to understand why their simulation was not returning expected results.