{1479} revision 6 modified: 09-24-2019 02:25 gmt

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=σLN A = \sigma L N AA = absorbance; LL = sample length, 10 μm, 1e-3 cm; NN = concentration, 10 μmol; σ\sigma = cross-section, for ETPA assume 2.4e18cm 2/molec2.4e-18 cm^2 / molec (this is based on a FMN based fluorophore; actual cross-section may be higher). Including Avogadro's number and 1l=1000cm 31 l = 1000 cm^3 , A=1.45e5A = 1.45e-5

Now, add in quantum efficiency ϕ=0.8\phi = 0.8 (Rhodamine); collection efficiency η=0.2\eta = 0.2 ; and an incoming photon pair flux of I=1e12photons/sec/modeI = 1e12 photons / sec / mode (which roughly about the limit for quantum behavior; n = 0.1 photons / mode; will add this calculation).

F=ϕησLNI=2.3e6photons/secF = \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 7e57e5 addressable modes in the FOV. Then an illumination of 1e121e12 photons / sec / mode equates to 7e177e17 photons over the whole field; if each photon pair has an energy of 2.75eV,λ=450nm2.75 eV, \lambda = 450 nm , this is equivalent to 300 mW. 100mW is a reasonable limit, hence scale incoming flux to 2.3e172.3e17 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, 2.3e172.3e17 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.