B Limit of Focal Spot Size for Lensing EM Radiation Across Frequencies

[houlahound]

Did not know how to word this properly.

Looking for an equation to show the behaviour of how EM can be "lensed" as a function of wavelength/frequency.

ie RF waves can not be lensed/focussed to a spot.

what equation determines the minimum spot size a lens can focus EM radiation as a function of frequency?

 

[Simon Bridge]

No EM waves can be focussed down to a point ... the narrowest part of a beam of converging rays is called the "beam waist".
https://en.wikipedia.org/wiki/Gaussian_beam#Beam_waist

BTW: it is possible to build a converging lens for radio waves.
iirc paraffin wax used to be used ... there is also a lune lens, and more recently they have got good
http://news.mit.edu/2012/new-metamaterial-lens-focuses-radio-waves-1114

 

[Tom.G]

Is this what you are asking about? The Abbe diffraction limit of a microscope, which is r = λ/2.8. Shine a light thru the eyepiece and that is the minimum obtainable spot radius.
https://en.wikipedia.org/wiki/Diffraction-limited_system

 

[Drakkith]

what equation determines the minimum spot size a lens can focus EM radiation as a function of frequency?

I believe you're asking about the Airy Disk. Try the math in this article: https://en.wikipedia.org/wiki/Airy_disk

 

[houlahound]

thanks guys. this is precisely the physics I was looking for.

haven't read through it all in detail but am I right in thinking the spot size is purely a function of the hardware transforming the light?

in the airy pattern, which is what is caused by diffraction from a circular aperture, would that increase the beam diameter after the aperture ie in general the smaller the aperture will cause greater spreading ie more diffraction. so therefore, if I am thinking right, a smaller beam will be achieved by a transparent lens than an aperture via diffraction.

I am going to try link the beam waist formula to the airy disc radius, but not sure if I am messing up the wrong two concepts.

 

[Drakkith]

in the airy pattern, which is what is caused by diffraction from a circular aperture, would that increase the beam diameter after the aperture ie in general the smaller the aperture will cause greater spreading ie more diffraction. so therefore, if I am thinking right, a smaller beam will be achieved by a transparent lens than an aperture via diffraction.

You're right in that a lens will make a smaller airy disk than an open aperture, but there's still a fundamental limit to how small the airy disk can be which is based on the size of the aperture and the wavelength of the incoming light/radiation. A larger aperture and/or smaller wavelength gives a smaller airy disk.