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= Photon Density Wave Spectroscopy = Photon Density Wave (PDW) Spectroscopy is an analytical technique used to determine the absorption and scattering properties of turbid dispersions. The technique has been also called diffused photon density wave spectroscopy and photon migration, with the same phenomenon being explored. The technique makes use of two optical fibres, one of which is responsible for the emission of photons and another that acts as a detector, with the emitted photons modulated in amplitude over time and the distance between fibres varied over time. Through analysis of changes in amplitude detected and the phase shift of the waves passing through the medium that reach the detector, the absorption and scattering effects of the solution can be independently evaluated. The scattering parameters can in turn, through Mie Theory, be converted to particle sizes. This particular capability is put to use for inline monitoring of particle sizes in concentrated dispersions, typically of use for industrial processes.

Setup
PDW Spectroscopy systems can be set up as either a two fiber type system where the distance between the fibres can be adjusted by a translation stage or other mechanically controlled system, or via a probe-type set up. In a probe configuration multiple fibers are arranged at varying distances from each other, and by changing which fibre is used as the emitter or detector different distances can be examined. The fibres themselves are connected to a fibre switch, which can switch the light source to different laser diodes set to different wavelengths, or switch to a network analyzer to set the fibre as a detector.

Theoretical Basis
The propagation of radiation is described through radiative transfer. The time dependent intensity $$\psi({\bf r},{\bf\Omega},t)$$ describes the number of photons per volume at position $${\bf r}$$ propagating in the direction of $${\bf\Omega}$$. Starting from the Boltzmann transport equation, the intensity field is described by the differential equation

$$ \left[{\bf\Omega}\cdot{\bf\nabla}+\frac{\partial}{c\partial t}+[\mu_a+\mu_s]\right]\psi({\bf r},{\bf\Omega},t) = \frac{1}{c}S({\bf r},{\bf\Omega},t)+\mu_s\int\psi({\bf r},{\bf\Omega'},t)f({\bf\Omega'},{\bf\Omega})d\Omega' $$

Where
 * $${\bf\nabla}$$ is the nabla operator
 * $$c$$ the speed of light in the medium
 * $$\mu_a$$ the absorption coefficient
 * $$\mu_s$$ the scattering coefficient
 * $$S({\bf r},{\bf\Omega},t)$$ describes the source
 * $$f({\bf\Omega'},{\bf\Omega})$$ is the scattering phase function and describes the probability of scattering towards $${\bf\Omega}$$ from $${\bf\Omega'}$$.

The differential equation is usually solved by developing the direction component of the terms in spherical harmonics and truncating the higher orders of the series. In the P1-approximation of the intensity, the spherical harmonic approximation of the intensity is

$$ \psi({\bf r},{\bf\Omega},t) \approx \sqrt{\frac{1}{4\pi}}\psi_0^0({\bf r},t)+\sqrt{\frac{3}{8\pi}} \begin{pmatrix} \psi_1^{-1}({\bf r},t)-\psi_1^1({\bf r},t)\\ -i\left(\psi_1^{-1}({\bf r},t)+\psi_1^1({\bf r},t)\right)\\ \sqrt{2}\psi_1^0({\bf r},t) \end{pmatrix}\cdot {\bf\Omega} $$

Through definition of the photon density $$\rho$$ and the photon flux density $${\bf J}$$, this simplifies to

$$ \psi({\bf r},{\bf\Omega},t) \approx \frac{\rho({\bf r},t)}{4\pi} + \frac{3{\bf J}({\bf r},t)\cdot {\bf \Omega}}{4\pi c} $$

With

$$ \rho({\bf r},t) = \int\psi({\bf r},{\bf\Omega},t)d{\bf \Omega} \approx \sqrt{4\pi}\psi_0^0({\bf r},t) $$

and

$$ {\bf J}({\bf r},t) = c\int\Omega\cdot\psi({\bf r},{\bf\Omega},t)d{\bf \Omega} \approx \sqrt{\frac{2\pi}{3}}c \begin{pmatrix} \psi_1^{-1}({\bf r},t)-\psi_1^1({\bf r},t)\\ -i\left(\psi_1^{-1}({\bf r},t)+\psi_1^1({\bf r},t)\right)\\ \sqrt{2}\psi_1^0({\bf r},t) \end{pmatrix} $$

Higher order approximations (P3, P5) have been described in the literature.

The source can be approximated as a uniform point source,

$$ S({\bf r},{\bf\Omega},t)\approx \frac{1}{4\pi} s_0({\bf r},t) $$

When neglecting polarization and interference, the scattering phase function can be approximated using the anisotropy factor g,

$$ f({\bf\Omega'},{\bf\Omega})\approx \frac{1+3g[{\bf\Omega'}\cdot{\bf\Omega}]}{4\pi} $$

Where g is definded as

$$ g=\frac{\int_{-1}^1 f({\bf\Omega'},{\bf\Omega})\cdot [{\bf\Omega'}\cdot{\bf\Omega}] d{\bf\Omega'}} {\int_{-1}^1 f({\bf\Omega'},{\bf\Omega}) d{\bf\Omega'}} $$

The above approximations applied to the differential equation result in

$$ \left[{\bf\Omega}\cdot {\bf\nabla}+ \frac{\partial}{c\partial t} + \left[\mu_a+\mu_s\right]\right] \left[\frac{\rho({\bf r},t)}{4\pi} + \frac{3{\bf J}({\bf r},t)\cdot {\bf \Omega}}{4\pi c}\right] = \frac{1}{4\pi c} s_0({\bf r},t)+ \mu_s\int \left[\frac{\rho({\bf r},t)}{4\pi} + \frac{3{\bf J}({\bf r},t)\cdot {\bf \Omega'}}{4\pi c}\right] \left[\frac{1+3g[{\bf\Omega'}\cdot{\bf\Omega}]}{4\pi}\right]d{\bf \Omega'} $$

Which simplifies to

$$ \left[{\bf\Omega}\cdot {\bf\nabla}+ \frac{\partial}{c\partial t} + \mu_a\right]c\rho({\bf r},t) + 3\left[ {\bf\Omega}\cdot {\bf\nabla}+ \frac{\partial}{c\partial t} + \mu_a + \mu_s' \right] {\bf J}({\bf r},t)\cdot {\bf \Omega} = s_0({\bf r},t) $$

Where $$\mu_s'=\mu_s [1-g]$$ is the reduced scattering coefficient. From this differential equation it is possible to derive a differential equation for the photon density $$\rho$$, by comparing two integrations. The first is obtained by integrating over $${\bf\Omega}$$, yielding

$$ {\bf\nabla}\cdot{\bf J}({\bf r},t) = s_0({\bf r},t) - \left[\frac{\partial}{c\partial t} + \mu_a\right]c\rho({\bf r},t) $$

The second equation is obtained by multiplying with $${\bf\Omega}$$, then integrating over $${\bf\Omega}$$ and finally multiplying with $${\bf\nabla}$$, yielding

$$ \frac{c}{3}{\bf\nabla}^2 \rho({\bf r},t) + \left[\frac{\partial}{c\partial t} + \mu_a + \mu_s' \right] {\bf\nabla} \cdot {\bf J}({\bf r},t) = 0 $$

Through insertion of $${\bf\nabla} \cdot {\bf J}({\bf r},t)$$ in the second equation the following is obtained

$$ \frac{c}{3}{\bf\nabla}^2 \rho({\bf r},t) + \left[\frac{\partial}{c\partial t} + \mu_a + \mu_s' \right] \left[s_0({\bf r},t) - \left[\frac{\partial}{c\partial t} + \mu_a\right]c\rho({\bf r},t)\right] = 0 $$

And after rearranging,

$$ \left[ \frac{c}{3\left[\mu_a + \mu_s' \right]}\frac{3\partial^2}{c^2\partial t^2} + \left[c+ \frac{c}{3\left[\mu_a + \mu_s' \right]}3\mu_a \right]\frac{\partial}{c\partial t} + c\mu_a - \frac{c}{3\left[\mu_a + \mu_s' \right]}{\bf\nabla}^2 \right]\rho({\bf r},t) = \frac{c}{3\left[\mu_a + \mu_s' \right]}\frac{3\partial}{c^2\partial t} s_0({\bf r},t) + s_0({\bf r},t) $$

At which point the optical diffusion coefficient $$D=\frac{c}{3\left[\mu_a + \mu_s' \right]}$$ is typically inserted,

$$ \left[ 3D\frac{\partial^2}{c^2\partial t^2} + \left[c+ 3D\mu_a \right]\frac{\partial}{c\partial t} + c\mu_a - D{\bf\nabla}^2 \right]\rho({\bf r},t) = \frac{3D}{c}\frac{\partial}{c\partial t} s_0({\bf r},t) + s_0({\bf r},t) $$

Solution for intensity modulated (sinusoidal) point light source
The above equation has been solved for specific circumstances. Here an infinite medium is considered with $$\mu_a<<\mu_s'$$. An intensity modulated isotropic point source at the origin has a constant amplitude $$q_{DC}$$ and a modulated amplitude $$q_{AC}$$ with frequency $$\omega$$.

$$ s_0({\bf r}, t) = \delta({\bf r})\left[q_{AC}\exp(-i\omega t)+q_{DC}\right] $$

Then the photon density fulfilling the conditions is

$$ \rho(r,t)=\frac{\rho_{AC}^{0}}{r}\exp(-k_I r + ik_\Phi r -i\omega t) + \frac{\rho_{DC}^{0}}{r}\exp(-k_{DC} r) $$

Where the spherical symmetry of the problem reduced the vector $${\bf r}$$ to the scalar distance to the origin $$r$$. The definitions of the parameters vary slightly depending on the approximations taken. Using the P1 approximation yields the parameters $$ \rho_{AC}^{0}=\frac{3q_{AC}}{4\pi c}\left[\alpha\mu_a+\mu_s'-\frac{i\omega}{c}\right] $$

$$ k_I = \sqrt{ \frac{3}{2}\sqrt{\left[\left[\alpha\mu_a+\mu_s'\right]^2+\omega^2 c^{-2}\right]\left[\mu_a^2+\omega^2 c^{-2}\right]} +\frac{3}{2}\mu_a\left[\alpha\mu_a+\mu_s'\right] -\frac{3}{2}\omega^2 c^{-2} } $$

$$ k_\Phi = \sqrt{ \frac{3}{2}\sqrt{\left[\left[\alpha\mu_a+\mu_s'\right]^2+\omega^2 c^{-2}\right]\left[\mu_a^2+\omega^2 c^{-2}\right]} -\frac{3}{2}\mu_a\left[\alpha\mu_a+\mu_s'\right] +\frac{3}{2}\omega^2 c^{-2} } $$

$$ \rho_{DC}^0 = \frac{q_{DC}}{4\pi D} $$

$$ k_{DC}=\sqrt{\frac{c\mu_a}{D}} $$

With $$\alpha=\frac{1}{3}$$. Conversely, the material parameters $$\mu_a$$ and $$\mu_s'$$ can be obtained from measurements,

$$ \mu_a = \frac {ck_I k_\Phi - \sqrt{c^2 k_I^2 k_\Phi^2 - 3\omega^2 \left[k_I^2 - k_\Phi^2 + 3\omega^2 c^{-2}\right]}} {3\omega} $$

$$ \mu_s' = \frac {\left[1-\alpha\right]c k_I k_\Phi + \left[1+\alpha\right]\sqrt{c^2 k_I^2 k_\Phi^2 - 3\omega^2 \left[k_I^2 - k_\Phi^2 + 3\omega^2 c^{-2}\right]}} {3\omega} $$

Data Analysis
In a typical PDW experiment, the intensity $$I$$ and phase offset $$\Phi$$ are measured for multiple frequencies $$\omega$$ and distances $$r$$ to the emitter. According to the equations above, the measurands follow the theoretical curves as

$$ I(\omega, r)~\propto \frac{\rho_{AC}^0(\omega)}{r} \exp(-k_I(\omega)r) $$

$$ \Phi(\omega,r) = k_\Phi (\omega)r+\Phi_0 $$

Through measuring $$I$$ and $$\Phi$$ for many distances and frequencies, the parameters can be fitted and $$\mu_a$$ and $$\mu_s'$$ determined.

Applications
PDW spectroscopy is used to determine absorption and scattering properties of optically dense systems, which are in themselves useful in some processes, but typically the desired output is particle size information (average size and distribution). The principle benefit of PDW spectroscopy is the ability to obtain meaningful data from concentrated suspensions where typical light scattering methods are no longer applicable due to multiple scattering events from short free path lengths for transitting photons. This means that in order to analyze concentrated suspensions with light scattering techniques samples must be taken offline and diluted to low enough concentrations (typically <1% by weight of sample) that representative data can be obtained. The process of dilution can affect the size and properties of particles desired to be examined and also introduces a time lag between sample collection and the analysis thereof, in which time particles can change in shape and morphology. Correspondingly, in highly dilute solutions the scattering events happen at larger scales, and often meaningful measurements cannot be made. As the photons reaching the detector need to be sufficiently scattered to give accurate information of the system, in addition to a lower limit of turbidity there also needs to be sufficient volume of the dispersion, which results in the technique being more applicable to larger scale systems. PDW spectroscopy is applied in analyzing:
 * polymer suspensions
 * food and drink production processes
 * Inorganic systems like titanium dioxide dispersions
 * Algae cultivation
 * Fermentation processes
 * Crystallization
 * Colloidal stability studies
 * and others.