How should the optical diffraction field be solved?

[TJULICHEN]

TL;DR

In Fourier optics analysis, is it reasonable to multiply the wavefront function by the screen function of the diffraction screen to obtain the wavefront distribution behind the screen? This approach seems to yield results that differ from those obtained using the Kirchhoff diffraction integral formula, which includes a tilt factor and the coefficient term -i/λ. However, in Fourier optics, it seems that such terms are not introduced. I am quite confused about this and it makes me feel that the tw

In Fourier optics analysis, is it reasonable to multiply the wavefront function by the screen function of the diffraction screen to obtain the wavefront distribution behind the screen? This approach seems to yield results that differ from those obtained using the Kirchhoff diffraction integral formula, which includes a tilt factor and the coefficient term -i/λ. However, in Fourier optics, it seems that such terms are not introduced. I am quite confused about this and it makes me feel that the two theories are not consistent. I would appreciate any expert guidance on this.

 

[Ibix]

Fourier optics is a far-field approximation, assuming that the source is small. It can be derived from the more general approach with the appropriate approximations. So they're compatible in the same sense that $\theta\approx\sin(\theta)$.

I would expect most books covering Fourier optics to cover the maths of this.

 

[TJULICHEN]

Fourier optics is a far-field approximation, assuming that the source is small. It can be derived from the more general approach with the appropriate approximations. So they're compatible in the same sense that $\theta\approx\sin(\theta)$.

I would expect most books covering Fourier optics to cover the maths of this.

Thank you very much for your answer; i.e., to calculate the diffracted field accurately, should I still use the Kirchhoff integral solution? Unfortunately, in most books deriving Fourier optical elements, for example, deriving the quadratic phase factor function of a lens, is based on the calculation of the optical range difference of a point source and does not explain the differences and approximations operating in the theory of diffraction with Kirchhoff. Anyway, thank you very much for your reply.

 

[Ibix]

This Wikipedia article derives Fresnel diffraction from EM theory. If I recall correctly, where the article drops the third and later terms in the Taylor series to get Fresnel, you get Fraunhofer diffraction by dropping the second term as well. It includes a textbook reference, and I'd be a little surprised if Born and Wolf didn't cover it.

 

[Andy Resnick]

Thank you very much for your answer; i.e., to calculate the diffracted field accurately, should I still use the Kirchhoff integral solution? [...]

Just to add to @Ibix's well-considered comments:

It depends on what you mean by "accurately". Yes, the Kirchoff diffraction integral is valid over a larger range of length scales than Fraunhofer's, but the Kirchoff boundary conditions are inconsistent. To overcome that, use Rayleigh-Sommerfeld diffraction theory.

Because the Rayleigh-Sommerfeld integrals (there are 2 of them) are difficult to evaluate, a variety of approximations are used: the Fresnel approximation and Fraunhofer approximation are both paraxial approximations while the Debye approximation is appropriate in the vicinity of a focal region of high numerical aperture lenses and involves an additional set of apodization functions (sine condition, Helmholtz condition, Herschel condition, etc...) to parameterize the lens aperture.

However, all of these various formulations and approximations only apply to *scalar* diffraction. Vector diffraction theory, AFAIK, has only been examined in the case of the Debye approximation.

The essential references (IMO) for all this are the usual suspects: Born and Wolf "Principles of Optics", Goodman "Introduction to Fourier Optics", and Gu "Advanced Optical Imaging Theory".

 

[TJULICHEN]

This Wikipedia article derives Fresnel diffraction from EM theory. If I recall correctly, where the article drops the third and later terms in the Taylor series to get Fresnel, you get Fraunhofer diffraction by dropping the second term as well. It includes a textbook reference, and I'd be a little surprised if Born and Wolf didn't cover it.

"Thank you very much for your answer. I now have a general understanding."

 

[TJULICHEN]

Just to add to @Ibix's well-considered comments:

It depends on what you mean by "accurately". Yes, the Kirchoff diffraction integral is valid over a larger range of length scales than Fraunhofer's, but the Kirchoff boundary conditions are inconsistent. To overcome that, use Rayleigh-Sommerfeld diffraction theory.

Because the Rayleigh-Sommerfeld integrals (there are 2 of them) are difficult to evaluate, a variety of approximations are used: the Fresnel approximation and Fraunhofer approximation are both paraxial approximations while the Debye approximation is appropriate in the vicinity of a focal region of high numerical aperture lenses and involves an additional set of apodization functions (sine condition, Helmholtz condition, Herschel condition, etc...) to parameterize the lens aperture.

However, all of these various formulations and approximations only apply to *scalar* diffraction. Vector diffraction theory, AFAIK, has only been examined in the case of the Debye approximation.

The essential references (IMO) for all this are the usual suspects: Born and Wolf "Principles of Optics", Goodman "Introduction to Fourier Optics", and Gu "Advanced Optical Imaging Theory".

Thank you for your answer; it has been very helpful！

 

[TJULICHEN]

This Wikipedia article derives Fresnel diffraction from EM theory. If I recall correctly, where the article drops the third and later terms in the Taylor series to get Fresnel, you get Fraunhofer diffraction by dropping the second term as well. It includes a textbook reference, and I'd be a little surprised if Born and Wolf didn't cover it.

Just to add to @Ibix's well-considered comments:

It depends on what you mean by "accurately". Yes, the Kirchoff diffraction integral is valid over a larger range of length scales than Fraunhofer's, but the Kirchoff boundary conditions are inconsistent. To overcome that, use Rayleigh-Sommerfeld diffraction theory.

Because the Rayleigh-Sommerfeld integrals (there are 2 of them) are difficult to evaluate, a variety of approximations are used: the Fresnel approximation and Fraunhofer approximation are both paraxial approximations while the Debye approximation is appropriate in the vicinity of a focal region of high numerical aperture lenses and involves an additional set of apodization functions (sine condition, Helmholtz condition, Herschel condition, etc...) to parameterize the lens aperture.

However, all of these various formulations and approximations only apply to *scalar* diffraction. Vector diffraction theory, AFAIK, has only been examined in the case of the Debye approximation.

The essential references (IMO) for all this are the usual suspects: Born and Wolf "Principles of Optics", Goodman "Introduction to Fourier Optics", and Gu "Advanced Optical Imaging Theory".

Please forgive me, as I am still confused about one issue and hope you can help me clarify it. Suppose I have several point sources of spherical waves (treated as scalar waves) distributed in the XOZ plane, with no other objects in the space. When calculating the phase distribution on a receiving surface, should I use the interference concept: by calculating the wavefront of each point source on the receiving surface and then performing coherent superposition? Or should I solve it using the Kirchhoff integral formula on the XOZ plane? Will the results be the same for both approaches?

 

[Ibix]

They should give the same answer. I would expect the interference method to be easier, because the Kirchhoff method does all the extra work of deriving the solution to each point source's emission whereas the interference method starts with the solution to an individual point source and adds them.

If you're having trouble, can you derive the single point source solution from Kirchhoff for an arbitrarily located source? That should then show you that adding discrete point sources by either method is the same.