# Effect of lens diameter on spot size

I would like to know if the lens diameter, i.e the size of the lens, affect the focused spot size. Given a parallel beam of diameter d, is there a lower limit of lens diameter D, after which aberration effects, simple linear ray-tracing fails, and the spot is no longer the smallest size possible?

I am participating in a competition which requires focusing laser beam. Since the beam diameter of diode lasers are small, I wish to know if I could use lens of size 10s of orders of magnitude of the beam diameter to focus it to a small spot.

• As in your related question, I think that the dimension of the lens can (or cannot, if big enough) affect the focus of the spot. If I remember correctly, in "Introduction to Fourier Optics" by Goodman should be a chapter about diffraction and this kind of effect. Dec 23, 2016 at 8:52
• Possible duplicate: physics.stackexchange.com/q/41597/24774, Dec 23, 2016 at 11:43

The potential spotsize at the focus of an imaging system varies inversely with the numerical aperture; there are various equations that express this relationship, depending on how you measure the spotsize. The commonest is (but see below for others):

$$\mu \approx \frac{0.61\,\lambda}{\eta}\tag{1}$$

where $\eta$ is the numerical aperture, which is the sine of half the angle subtended by the lens diameter at the focus.

However: if you do try to reproduce this relationship with a set of single, biconvex spherical lenses, it will only work for numerical apertures up to about $\eta\approx0.07$. A spherical surface lens does not focus rays to a point - it only does so approximately for small numerical apertures and above this $\eta\approx0.07$ "speed" of the lens, the lens introduces so much wavefront error to the wavefront that the focus is spoilt, the system no longer works in its diffraction limited mode and the spotsize will thus begin to increase with increasing lens size.

If you want to reach the resolution implied by (1), you must either build a multi-element lens group to correct the optical aberration or you must use aspherical lenses, which are shapen to focus rays perfectly to point.

My favorite way of stating the relationship is:

$$\mu \geq \frac{2\,\lambda}{\pi\,\eta}\tag{2}$$

where $\eta$ is the numerical aperture, $\lambda$ the wavelength and $\mu$ the spotsize; both $\mu$ and $\eta$ are measured as $2\sqrt{2}$ times the intensity weighted second moment of distance / wavenumber distance from the optical axis / optical axis direction for spotsizes and numerical aperture, respectively. These are the Petermann II method of defining these quantities, being the intensity weighted second moment spread measured in optical and Fourier space. In the aberration free case, the inequality is saturated for Gaussian apodized fields and indeed (2) is an analogy of the Heisenberg inequality. But this analogy only arises because the mathematics of Fourier optics is the same as the mathematics that shows the position and momentum co-ordinate wavefunctions to be Fourier transforms of one another in quantum mechanics. (2), although a mathematical precise analogy, has nothing to do with the quantum mechanics of light or the Heisenberg uncertainty principle. The reason I like these definitions is because second moment definitions and (2) is that they take account of the apodisation of a focussing optical field.