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How can I simulate PSF of incoherent light? 
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#1
Feb1913, 02:18 AM

P: 15

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 


#2
Feb1913, 04:29 AM

Sci Advisor
P: 3,628

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



#3
Feb1913, 07:37 AM

P: 15

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 xlinear and ylinear. 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 


#4
Feb1913, 05:34 PM

Sci Advisor
P: 5,539

How can I simulate PSF of incoherent light?
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. 


#5
Feb2013, 01:39 AM

P: 15

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 gonna read the book 'Advanced optical imaging theory'. Regards. 


#6
Feb2013, 07:11 AM

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P: 5,539

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 6.5.96.5.13); 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, 325332 (1995). P. Török and P. Varga, "Electromagnetic diffraction of light focused through a stratified medium," Appl. Opt. 36, 23052312 (1997). 


#7
Feb2013, 08:09 AM

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P: 3,628

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_Cit...ernike_theorem 


#8
Feb2013, 09:24 AM

P: 15

I do appreciate your timely help Best regards 


#9
Feb2113, 07:37 AM

P: 15

I gonna try this method. Regards 


#10
Feb2113, 08:35 AM

Sci Advisor
P: 5,539

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(1cosθ)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 + 4I_1^2 cos^2(ψ)+I_2^2 +2cos(2ψ)Re(I_0 I_2*)] Does this help? 


#11
Feb2213, 07:09 AM

P: 15

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. 


#12
Feb2513, 08:30 AM

Sci Advisor
P: 5,539

Also, your [2] looks vaguely familiar where did that come from? Are you able to recover 'known' results if your NA is small? 


#13
Feb2513, 09:28 AM

P: 15

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. 


#14
Feb2613, 07:17 PM

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P: 5,539

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


#15
Feb2713, 04:14 AM

P: 15

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 xlinear and ylinear 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 


#16
Feb2813, 07:33 AM

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#17
Feb2813, 10:23 AM

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P: 3,628

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? 


#18
Mar113, 02:33 AM

P: 15

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 


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