**Can we image biological tissue with entangled photons?**
How much fluorescence can we expect, based on reasonable concentrations & published ETPA cross sections?
Start with beer's law: $A = \sigma L N$ $A$ = absorbance; $L$ = sample length, 10 μm, 1e-3 cm; $N$ = concentration, 10 μmol; $\sigma$ = cross-section, for ETPA assume $2.4e-18 cm^2 / molec$ (this is based on a FMN based fluorophore; actual cross-section may be higher). Including Avogadro's number and $1 l = 1000 cm^3$ , $A = 1.45e-5$
Now, add in quantum efficiency $\phi = 0.8$ (Rhodamine); collection efficiency $\eta = 0.2$ ; and an incoming photon *pair* flux of $I = 1e12 photons / sec / mode$ (which roughly about the limit for quantum behavior; n = 0.1 photons / mode; will add this calculation).
$F = \phi \eta \sigma L N I = 2.3e6 photons/sec$ This is very low, but within practical imaging limits. As a comparison, incoherent 2p imaging creates ~ 100 photons per pulse, of which 10 make it to the detector; for 512 x 512 pixels at 15fps, the dwell time on each pixel is 20 pulses of a 80 MHz Ti:Sapphire laser, or ~ 200 photons.
Note the *pair* flux is per optical mode; for a typical application, we'll use a Nikon 16x objective with a 600 μm Ø FOV and 0.8 NA. At 800 nm imaging wavelength, the diffraction limit is 0.5 μm. This equates to about $7e5$ addressable modes in the FOV. Then an illumination of $1e12$ photons / sec / mode equates to $7e17$ photons over the whole field; if each photon pair has an energy of $2.75 eV, \lambda = 450 nm$ , this is equivalent to 300 mW. 100mW is a reasonable limit, hence scale incoming flux to $2.3e17$ pairs /sec.
Hence, the imaging mode is power limited, and not quantum limited (**if** you could get such a bright entangled source). And right now that's the limit -- for a BBO crystal, circa 1998 experimenters were getting 1e4 photons / sec / mW. So, $2.3e17$ pairs / sec would require *23 GW*. Yikes.
More efficient entangled sources have been developed, using *periodically-poled potassium titanyl phosphate* (PPPTP), which (again assuming linearity) puts the power requirement at 23 MW. This is within the reason of q-switched lasers, but still incredibly inefficient. The down-conversion process is not linear in intensity, which is why Goodson pumps with SHG from a Ti:sapphire to yield ~1e7 photons; but this of induces temporal correlations which increase the frequency of incoherent TPA.
Still, combining PPPTP with a Ti:sapphire laser could result in 1e13 photons / sec, which is sufficient for *scanned* microscopy. Since the laser is pulsed, it will still be subject to incoherent TPA; but that's OK, the point is to reduce the power going into the animal via larger ETPA cross-section. The answer to above is a tentative **yes**. Upon the development of brighter entangled sources (e.g. arrays of quantum structures), this can move to fully widefield imaging. |