# Calculation of diffraction picture (partially coherent illumination)

1. Sep 20, 2009

### Sergei_G

Hello,

I need to calculate diffraction picture produced by source generating partially coherent illumination. I read Born and Wolf, Goodman and still I do not have complete understanding.

Suppose we have source with size d, simple object (let's take opaque rectangle) and detector. According to illumination wavelength, object size and setup geometry image can be calculated in Fresnel approximation. How shall I calculate image considering that source produces partially coherent radiation with degree of coherency $$\gamma$$?

From what I have found: I can take wave field P(x,y)=P0(x,y)exp[ikW(x,y)] -
amplitude and phase of wave right after it passes through the object). I know Raleigh Sommerfeld RS propagator h(x,y). Than I can calculate image for point radiator:
I(x,y)=abs(F-1F(P)F(h))2 (F is fast fourier transform)
And take convolution of this value over all point radiatiors in the source. Is it case of coherent or incoherent illumination?

What about convolution of intensities
Iincoherent=|h|2convolved with|P|2 ( formula from Goodman, Fourier Optics sec 6.5.1) Shall I take squared modulus of wave field and RS (plain wave etc.) propagator and than calculate like this:
Iincoh(x,y)=F-1F(abs(h)2)F(abs(P)2) ??? How I have to take into account convolution over source size in this case?

Ipartially coherent=$$\gamma$$Icoh+(1-$$\gamma$$)Icoh ??? Can I use it?

I also found in Born Wolf that main value for the partially coherent illumination is Mutual Intensity, how can I calculate it and is it necessary?

Could you please clarify this for me?

Last edited: Sep 20, 2009
2. Sep 20, 2009

### Andy Resnick

Yikes...

I think you understand the limiting cases- fully coherent illumination and fully incoherent illumination. For the first, what is calculated is the field: an object point gets replaced by the far-field diffraction pattern of a point, which is the Fourier Transform of the field at the pupil plane- the spatial Fourier transform of the aperture (with the exp(ikW) part to model aberrations). In the second, what is calculated is the intensity- an object point is replaced with the square of the Fourier transform of the aperture. Alternatively, the MTF of a coherent system is the field at the pupil plane, while the MTF of an incoherent system is the autocorrelation of the field at the pupil plane.

For partially coherent light, I have to double-check what I think the result is. The size of the source gives a measure of the partial coherence (the transverse coherence length), but the mutual coherence function is a complex function of the field at (at least) two points. I'm not sure if I have seen a simplified, clean result for your geometry. Where did the formula 'Ipartially coherent=LaTeX Code: \\gamma Icoh+(1-LaTeX Code: \\gamma )Icoh' come from?

3. Sep 21, 2009

### Sergei_G

Thanks for the fast response.

I'm still not sure at all if I understand this properly ^^ More precisely, I don't understand, what is formula of incoherent case for, if I have point source? Spatial coherence relates to the finit source size, doesn't it? What does formula with convolution of intensities mean?

If I calculate intensity in image plane for point source using convolution of complex propagator h and wave field P and take it squared module - I will get picture with maximum visibility since radiation is completely spatially coherent. Than I make convolution of this picture over source size - shifted pictures are summurized and visibility decreases, because now radiation is not spatially coherent. So how convolution of intensities connected with this? Where is source size in this equation Iincoherent=|h|2convolved with|P|2? Or I still have to make convolution over source size after calculating image from point source?

Formula Ipartially coherent=gamma I+(1-gamma )I is from "Principles of Optics", 6th edition, sec. 10.3, "Correlation functions of light beams" p 502. I wonder, can I use it or shall I calculate image for partially coherent light using Mutual Intensity (table XXIV in sec 10.5 - The action of optical system from the standpoint of its response to spatial frequencies)

4. Sep 21, 2009

### Andy Resnick

Hmmm... Ok, I think there's too much here to handle all at once. Let's start with the difference between perfectly coherent and perfectly incoherent imaging.

If you have a point source, you cannot have (spatially) incoherent imaging, as you point out. "coherence" can refer to several different components of the system- the degree of coherence of the illumination, but also whether or not the *detector* can detect the phase of the field. In optics, it cannot ('optics' meaning visible light: phase-sensitive detection devices exist up to several GHz, IIRC)

So, if your detector can detect the phase, you have a coherent detector and since it is detecting the complex field amplitude, the image related to the (complex) Fourier transofrm of the aperture. For a phase-insensitive detector, since we only detect the intensity (i.e. the modulus squared of the field), the image is related to the modulus squared of the aperture. Goodman has a nice example comparing the two in sections 6-5 (and the two equations just above 6-39 in my edition).

Ok so far?

5. Sep 21, 2009

### Sergei_G

Mmm, let's consider exposure time to be much larger than coherence time.

I just checked again Fourier Optics, sec 6.1.3 Polychr Illumination, The Coh. and Incoh. Cases:
(1) "Thus while any two object points may have different rel-
ative phases, their absolute phases are varying with time in a perfectly correlated way.
Such illumination is called spatially coherent"+"Coherent illumination is obtained whenever light appears to originate from a single point"
(2) "...object illumination ... the phasor amplitudes at all points on the object are varying in totally uncorrelated fashions. Such illumination is called spatially incoherent".

So let me ask my question in other fashion:
(a) I calculate image from point source using wave field convolution equation hxP (let's x denotes convolution sign) and than convolve it over all radiators in the source.
(b) I calculate image using equation |h|^2 x |P|^2 (incoherent case according to books)
So if in the case (a) I increase source size to infinity, shall I get the same result as in the case (b)? (I will try to check it right now)

6. Sep 22, 2009

### Andy Resnick

I wonder if you are mixing the idea of 'coherent vs. incoherent' detection (i.e. amplitude or intensity) with 'coherent vs. incoherent' properties of the illumination field.

To answer (a), if I understand what you are writing, the image field *is* h x P. P is the field at the exit aperture. A source point gets mapped to h, and the image is h convolved with the object. The intensity image is |h x P|^2

For (b), the same holds- a source point gets mapped to |h|^2, which is convolved with |P|^2, so the intensity image is |h|^2 x|P|^2.

Note that has nothing to do with the coherence state of the illumination- in both cases, the illumination is considered fully coherent- the object is illuminated by a plane wave in both cases.

7. Dec 9, 2010

### Silviu_C

Sergei, it is called the Schell's theorem and it may be found at page 226 of Goodman book Statistical Optics. It says that the diffracted field is the FT of the pupil autocorrelation multiplied by the complex coherence factor miu.
hope it is the answer.