Photon-density waves

For an infinite homogeneous medium and a point source at the origin, one can write the frequency-domain energy density \((U_{FD})\), which is the fluence rate divided by the speed of light \((U_{FD}=\phi_{FD}/c)\), as follows (now explicitly including the sinusoidal factor \(e^{-i\omega t})\): 

\(U_{FD}\left(r,t,\omega\right)=\frac{P_{FD}\left(\omega\right)\ }{4\pi D}\frac{e^{-k_{Im}\left(\omega\right)r}}{r}e^{i\left[k_{Re}\left(\omega\right)r-\omega t\right]}\),                   (1) 

where \(k_{Re}\left(\omega\right)\) and \(k_{Im}\left(\omega\right)\) are the real and imaginary parts of a complex wavenumber, and are given by:

\(k_{Re}\left(\omega\right)=\frac{\mu_{eff}}{\sqrt2}\sqrt{\sqrt{1+\left(\frac{\omega}{c_n\mu_a}\right)^2}-1}\),                   (2) 

\(k_{Im}\left(\omega\right)=\frac{\mu_{eff}}{\sqrt2}\sqrt{\sqrt{1+\left(\frac{\omega}{c_n\mu_a}\right)^2}+1}\).                   (3) 

[Recall that \(\mu_{eff}=\sqrt{3\mu_a\left(\mu_s^\prime+\mu_a\right)}]\). The energy density of Eq. (1) has the mathematical form of a damped spherical wave propagating away from the light source. This spherical wave describes the temporal and spatial dependence of the energy density in a scattering medium. Since the energy density is directly related to the photon density, it is referred to as photon-density wave (PDW), for which the wavelength \((\lambda_{PDW})\), phase velocity \((v_{PDW})\), and attenuation length \((L_{PDW})\) are: 

\(\lambda_{PDW}=\frac{2\pi}{k_{Re}\left(\omega\right)}=\frac{2\pi}{\frac{\mu_{eff}}{\sqrt2}\sqrt{\sqrt{1+\left(\frac{\omega}{c_n\mu_a}\right)^2}-1}}\),                   (4) 

\(v_{PDW}=\frac{\omega}{k_{Re}\left(\omega\right)}=\frac{\omega}{\frac{\mu_{eff}}{\sqrt2}\sqrt{\sqrt{1+\left(\frac{\omega}{c_n\mu_a}\right)^2}-1}}\),                   (5) 

\(L_{PDW}=\frac{1}{k_{Im}\left(\omega\right)}=\frac{1}{\frac{\mu_{eff}}{\sqrt2}\sqrt{\sqrt{1+\left(\frac{\omega}{c_n\mu_a}\right)^2}+1}}\).                   (6) 

(It is important not to confuse \(\lambda_{PDW}\) and \(v_{PDW}\) with the optical wavelength and phase velocity of the light itself.) While photon-density waves are described by the same mathematical form as general damped waves, and in fact undergo typical wave phenomena such as reflection, refraction, and diffraction, they feature two properties that distinguish them from typical optical or ultrasound waves: 

1. They are strongly damped over distances short compared to their wavelength. Photon-density waves are attenuated by many orders of magnitude over one wavelength, so that they are observable only over a fraction of the wavelength, rendering them analogous to near-field waves;  
    
2. They are energy waves. Their amplitude relates linearly to an energy, as opposed to more common cases in which the energy associated with the wave is related to the square of the amplitude of oscillating quantities (for example, the electric field for electromagnetic waves or the pressure for ultrasound waves). 

The wave properties (wavelength and phase velocity) and the characteristic attenuation length of the PDW depend on the optical properties of the medium and on the modulation frequency of the source power. By considering typical values for frequency-domain spectroscopy of tissue, say \(\mu_a\) = 0.1 cm^-1, \(\mu_s^\prime\) = 10 cm^-1, n = 1.35, and f=\(\omega/\left(2\pi\right)\)= 100 MHz, one obtains representative values of \(\lambda_{PDW}\)~25 cm, \(v_{PDW}\)~c_n/10, and \(L_{PDW}\)~0.6 cm.