# How far can diffuse photons from a laterally focused laser beam penetrate in the axial direction?

I have a question regarding practical laser focusing in the axial direction. More specifically, I was looking to get an idea of how large a distance in the axial direction a laser focused at 200nm lateral spot size can penetrate in a biological specimen. To put this in context, I am interested in this question in the context of photoacoustic imaging. I am looking to gain some insight into the photoacoustic excitation volume of a focused laser beam. i.e. how deep in the z direction can we still generate a photoacoustic signal.

I have found the following equations which seem to be relevant $$\Delta x=\Delta y=\frac{\lambda}{2NA}$$ Where $\Delta x$ and $\Delta$ y are the lateral spot size parameters in x and y, respectively, $\lambda$ is the laser wavelength, and $NA$ is the numerical aperture. So I want to keep this number (for a 532nm laser) at about 200nm.

In addition to these, in the axial direction, I have found the equation $$\Delta z =\frac{2\lambda}{NA^2}$$ will put a limit on the distance that this focussed light stays focused before scattering in media.

However, to the best of my knowledge, these equations only apply to ballistic photons, and so are not necessarily descriptive of the total volume in which photoacoustic excitation will occur. In our experiments, we seek to maximize how far in the axial direction the photons (whether ballistic or diffuse) will penetrate while trying to maintain a lateral resolution of 200nm.

Are there any equations that describe how deep this light can penetrate? Specifically, is it physically possible for one to focus a laser to a spot size of 200nm while still having similar laser intensity as deep as 8um into a biological sample? We are wondering if it is theoretically possible to have a laser beam that is of high intensity in a (very very rough ) cylinder with diameter of 200nm and length of 8um.

Thanks in advance for any insight into this matter.

• Please be careful about your terms. Your second equation doesn't come from scattering, it comes from diffraction. But probably the answer to your question does depend on scattering phenomena, which will depend on properties of the "biological specimen". – The Photon May 15 '17 at 17:21
• Oops, yes that is accurate. I meant focusing not scattering. In this instance we can assume that the biological specimen is water for the sake of simplicity. Are there any equations that model the intensity distribution of diffuse photons in addition to ballistic ones? – hexagram May 15 '17 at 20:26
• Can you clarify what you mean by "ballistic photons"? The equations you cite are at least close to what you expect for classical light, with no dependence on the quantum nature (photons) at all. – The Photon May 15 '17 at 20:31
• I'm not quite a physicist so sorry for the sloppy terminology. I am mostly curious about the intensity distribution of a laterally focused laser beam after it has arrived on the surface of a sample. Here the intensity of the beam is the primary value of interest. – hexagram May 15 '17 at 21:09
• To be more specific, I am wondering if it is experimentally realistic to think that one may have a beam that penetrates 8um into the sample while still retaining a lateral spot size on the order of 200nm. Assuming a perfect Gaussian beam this should not be possible, nor should it be possible assuming a perfect diffraction limited beam. However, I was wondering if (due to out of focus light interference or some other phenomenon) such a geometry could possibly exist. – hexagram May 15 '17 at 21:10