absorption

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AbsorptionECE532 Biomedical Optics

©1998Steven L. Jacques, Scott A. Prahl

Oregon Graduate Institute

A simple analogy for the absorption of light by molecules is placing two identical bells side by side. When one bell is rung, the other will sympathetically ring at the same frequency due to transfer of energy from the struck bell. The resonance of the second bell matches the frequency of the first ringing bell and hence the second bell accepts energy from the struck bell.

Photons are electromagnetic waves with a particular frequency. Molecules are a system with charge separation (negative electron field and positive nucleus). The state of the molecular charge separation can change in a quantized fashion by "absorbing" the energy of a photon. The photon frequency must match the "frequency" associated with the molecule's energy transition in order for energy transfer to occur. The relation between frequency and energy is

Energy = h(frequency) = hc/(wavelength)

where Energy is in [J], photon frequency is in [cycles per s] or [s-1], photon wavelength is in [m], c is the speed of light in vacuo (c = 3.0x108 [m/s]), and h is Planck's constant (h = 6.62618x10-34 [J s]). Unlike the bell analogy, photon absorption occurs as a quantum event, an all or none phenomenon. Example: absorption of 514 nm photon from an argon ion laser by a hemoglobin molecule.

In biomedical optics, absorption of photons is a most important event:

Absorption is the primary event that allows a laser or other light source to cause a potentially therapeutic (or damaging) effect on a tissue. Without absorption, there is no energy transfer to the tissue and the tissue is left unaffected by the light.

Absorption of light provides a diagnostic role such as the spectroscopy of a tissue. Absorption can provide a clue as to the chemical composition of a tissue, and serve as a mechanism of optical contrast during imaging. Absorption is used for both spectroscopic and imaging applications.

Some biological chromophoresECE532 Biomedical Optics



Molecules that absorb light are called chromophores. There are two major types of choromphores:

electronic transitions vibrational transitions

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Electronic transitions

There are many biological molecules which can absorb light via electronic transitions. Such transitions are relatively energetic and hence are associated with absorption of ultraviolet, visible and near-infrared wavelengths. The molecules generally have a string of double bonds whose pi-orbital electrons act similar to the electrons in a metal in that they collectively behave as a small antenna which can "receive" the electromagnetic wave of a passing photon. If the resonance of the pi-orbital structure matches the photon's wavelength then photon absorption is possible.

In early biological evolution, the pyrrole molecule was a chromphore which could absorb sunlight which enabled subsequent synthetic reactions that produced biological polymers and other proto-metabolic products. Combining four pyrroles into a tetrapyrrole ring (porphyrin) yielded an efficient chromophore for collecting solar photons. Chlorophyll is such a porphyrin. Hemoglobin, vitamin B12, cytochrome C, and P450 are also examples of porphyrins in biology. The figure lists some common biological chromophores and shows some of their structures. Also see the website spectra which is a compilation of chromophores, most absorbing in the ultraviolet and visible.

Vibrational transitions

The field of infrared spectroscopy studies the variety of bonds which can resonantly vibrate or twist in response to infrared wavelengths and thereby absorb such photons. Perhaps the most dominant chromophore in biology which absorbs via vibrational transitions is water. In the infrared, the absorption of water is the strongest contributor to tissue absorption.

http://omlc.ogi.edu/spectra/

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Some vibrational frequencies

bond cycles/cm, wavelength, = 1/

C-H stretch 2850-2960 [cm-1] 3.378-3.509 [µm]

C-H bend 1340-1465 6.826-7.462

C-C stretch,bend 700-1250 8.000-14.29

C=C stretch 1620-1680 5.952-6.173

C=C stretch 2100-2260 4.425-4.762

CO32- 1410-1450 6.897-7.092

NO3- 1350-1420 7.042-7.407

NO2- 1230-1250 8.000-8.130

SO42- 1080-1130 8.850-9.259

O-H stretch 3590-3650 2.740-2.786

C=O stretch 1640-1780 5.618-6.098

N-H 3200-3500 2.857-3.125

ref: PW Atkins, "Physical Chemistry," p. 576, W.H. Freeman and Co., 1978.

Definition and units of absorption coefficient µa [cm-1]

ECE532 Biomedical Optics



Consider a chromophore idealized as a sphere with a particular geometrical size. Consider that this sphere blocks incident light and casts a shadow, which constitutes

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absorption. This description is of course an incorrect and schematicized version of the real situation. However, it does provide a simple concept which captures the essence of the absorption coefficient, the parameter we use to describe the effectiveness of absorption.

The size of the absorption shadow is called the effective cross-section ( a [cm2]) and can be smaller or larger than the geometrical size of the chromophore (A [cm2]), related by the proportionality constant called the absorption efficiency Qa [dimensionless]:

The absorption coefficient µa [cm-1] describes a medium containing many chromophores at a concentration described as a volume density a [cm3]. The absorption coefficient is essentially the cross-sectional area per unit volume of medium.

Experimentally, the units [cm-1] for µa are inverse length, such that the product µaL is dimensionless, where L [cm] is a photon's pathlength of travel through the medium. The probability of survival (or transmission T) of the photon after a pathlength L is:

This expression for survival holds true regardless of whether the photon path is a straight line or a highly tortuous path due to multiple scattering in an optically turbid medium.

Example: BilirubinECE532 Biomedical Optics



The bilirubin molecule is a chromophore encountered when a newborn infant suffers from "jaundice", a syndrome in which the skin presents a yellow color. Bilirubin is a breakdown product of hemoglobin. Often there is significant hemolysis of red blood

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cells during child birth contributing to a transient bilirubin load. Normally, bilirubin binds to the serum protein albumin and is carried to the liver where enzymes convert it into a water-soluble form which is removed from the blood into the bile. But in newborns who have not yet developed sufficient enzymes to accomplish this task, the bilirubin accumulates in the blood, exceeds the holding capacity of the albumin, and spills into the skin to give the yellow skin color, and into the brain to cause irreversible brain damage (kernicterus). See article on optically monitoring bilirubinemia.

Let's approximate the values of A, a, µa, Qa, and a for bilirubin.

The structure of bilirubin is shown. The diameter is approximately 1 nm. So the geometrical area is A = 4.5x10-15 cm2.

At 460 nm, the extinction coefficient of bilirubin is = 53,846 [cm-1M-1] (see bilirubin spectrum).

The extinction coefficient [cm-

1M-1] quoted in the literature is based on spectrometer measurements reported as T = 10-

CL where C is concentration [M] and L is pathlength [cm]. Therefore,

µa = C ln(10)

A typical jaundiced neonate might have a bilirubin concentration of 10 mg/dL, or (0.100 g/liter)/(574.65 g/mole) = 0.17x10-3 M. In such a case, the bilirubin absorption coefficient at 460 nm is roughly

µa = C ln(10) = (53846 [cm-1M-

1])(0.17x10-3 M)(2.3) = 21 cm-1.

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The concentration C is equivalent to

a = (0.17x10-3 [moles/liter])(6x1023

[mole-1])/(1000 cm3/liter] = 1.02x1017 [cm-3]

The efficiency of absorption is estimated:

Qa = µa/( aA) = (21 [cm-

1])/((1.02x1017 [cm-3])(A = 4.5x10-15 [cm2])) = 0.046

The effective cross-section is a = QaA = (0.046)(4.5x10-15 [cm2]) = 2.1x10-16 [cm2]. Bilirubin's effective cross-sectional diameter is sqrt(.046) or 21% the size of its geometrical diameter.

Keep in mind that the wavelength of blue light (460 nm) is about 460-fold greater than the diameter of the bilirubin chromophore. So collection of the electromagnetic wave of a photon by the "antenna" of bilirubin is analogous to a how small radio antenna collects the electromagnetic wave from a radio station (a 1000 MHz radio frequency -> 300 m wavelength). The concept of a "shadow" cast by an opaque chromophore is merely a memory device to remember the definitions.

Absorption spectra for biological tissuesECE532 Biomedical Optics



The figure below shows the primary absorption spectra of biological tissues. Also shown are the absorption coefficients at some typical laser wavelengths.

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There are several major contributors to the absorption spectrum:

In the ultraviolet, the absorption increases with shorter wavelength due to protein, DNA and other molecules.

In the infrared, the absorption increases with longer wavelengths due to tissue water content. Scaling the pure water absorption by 75% mimics a typical tissue with 75% water content.

In the red to near-infrared (NIR), absorption is miminal. This region is called the diagnostic and therapeutic window (originally by John Parrish and Rox Anderson).

Whole blood is a strong absorber in the red-NIR regime, but because the volume fraction of blood is a few percent in tissues, the average absorption coefficient that affects light transport is moderate. However, when photons strike a blood vessel they encounter the full strong absorption of whole blood. Hence, local absorption properties govern light-tissue interactions, and average absorption properties govern light transport.

Melanosomes are also strong absorbers. However, their volume fraction in the epidermis may be quite low, perhaps several percent. So again, the local interaction of light with the melanosomes is strong, but the melanosome contribution to the average absorption coefficient may modestly affect light transport.

ScatteringECE532 Biomedical Optics



Scattering of light occurs in media which contains fluctuations in the refractive index n, whether such fluctuations are discrete particles or more continuous variations in n.

In biomedical optics, scattering of photons is an important event:

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Scattering provides feedback during therapy. For example, during laser coagulation of tissues, the onset of scattering is an observable endpoint that correlates with a desired therapeutic goal. Scattering also strongly affects the dosimetry of light during therapeutic procedures that depend on absorption. The scattering affects "where" the absorption will occur.

Scattering has diagnostic value. Scattering depends on the ultrastructure of a tissue, eg., the density of lipid membranes in the cells, the size of nuclei, the presence of collagen fibers, the status of hydration in the tissue, etc. Whether one measures the wavelength dependence of scattering, the polarization dependence of scattering, the angular dependence of scattering, the scattering of coherent light, scattering measurements are an important diagnostic tool. Scattering is used for both spectroscopic and imaging applications

Some biological scatterers ECE532 Biomedical Optics



The light scattered by a tissue has interacted with the ultrastructure of the tissue. Tissue ultrastructure extends from membranes to membrane aggregates to collagen fibers to nuclei to cells. Photons are most strongly scattered by those structures whose size matches the photon wavelength.

Scattering of light by structures on the same size scale as the photon wavelength is described by Mie theory. Scattering of light by structures much smaller than the photon wavelength is called the Rayleigh limit of Mie scattering, or simply Rayleigh scattering. The figure designates the size range of tissue ultrastructure which affects visible and infrared light by Mie and Rayleigh scattering.

Here are some examples of structures which scatter light (click on figure to expand):

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Mitochondria

Mitochondria are intracellular organelles about 1 µm in length (variable) which are composed of many folded internal lipid membranes called cristae, as shown in the electron micrograph at left. The basic lipid bilayer membrane is about 9 nm in width. The refractive index mismatch between lipid and the surrounding aqueous medium causes strong scattering of light. Folding of lipid membranes presents larger size lipid structures which affect longer wavelengths of light. The density of lipid/water interfaces within the mitochondria make them especially strong scatterers of light.

(Drawing and micrographs from A. L. Lehninger,

"Biochemistry", Worth Publishers, 1970.)

Collagen fibers, fibrils, and fibril periodicity

Collagen fibers (about 2-3 µm in diameter) are composed of bundles of smaller collagen fibrils about 0.3 µm in diameter (variable), as shown in the electron micrograph at left. Mie scattering from collagen fibers dominates scattering in the infrared wavelength range.

On the ultrastructural level, fibrils are composed of entwined tropocollagen molecules. The fibrils present a banded pattern of striations with 70-nm periodicity due to the staggered alignment of the tropocollagen molecules which each have an electron-dense head group that appears dark in the electron micrograph. The periodic fluctuations in

http://omlc.ogi.edu/classroom/ece532/class3/gifs/mito.gif

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refractive index on this ultrastructureal level appear to contribute a Rayleigh scattering component that dominates the visible and ultraviolet wavelength ranges.

Definition and units of scattering coefficient

µs [cm-1]




Consider a scattering particle idealized as a sphere with a particular geometrical size. Consider that this sphere redirects incident photons into new directions and so prevents the forward on-axis transmission of photons, thereby casting a shadow. This process constitutes scattering. This description is of course an oversimplified and schematicized version of the real situation. However, it does provide a simple concept which captures the essence of the scattering coefficient, a parameter analogous to the absorption coefficient discussed previously.

The size of the scattering shadow is called the effective cross-section ( s [cm2]) and can be smaller or larger than the geometrical size of the scattering particle (A [cm2]), related by the proportionality constant called the scattering efficiency Qs [dimensionless]:

The scattering coefficient µs [cm-1] describes a medium containing many scattering particles at a concentration described as a volume density s [cm3]. The scattering coefficient is essentially the cross-sectional area per unit volume of medium.

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Experimentally, the units [cm-1] for µs are inverse length, such that the product µsL is dimensionless, where L [cm] is a photon's pathlength of travel through the medium. The probability of transmission T of the photon without redirection by scattering after a pathlength L is:

Definition of anisotropy

g [dimensionless]


©199813 Steven L. Jacques, Scott A. Prahl


The anistoropy, g [dimensionless], is a measure of the amount of forward direction retained after a single scattering event. Imagine that a photon is scattered by a particle so that its trajectory is deflected by a deflection angle , as shown in the figure below. Then the component of the new trajectory which is aligned in the forward direction is shown in red as cos( ). On average, there is an average deflection angle and the mean value of cos( ) is defined as the anisotropy. 13

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A scattering event causes a deflection at angle from the original forward trajectory. There is also an azimuthal angle of scattering, .

But it is the deflection angle which affects the amount of forward direction, cos( ), retained by the photon.

Consider an experiment in which a laser beam strikes a target such as a cylindrical cuvette containing a dilute solution of scattering particles. The scattering pattern p( ) is measured by a detector that is moved in a circle around the target while always facing the target. Hence the detector collects light scattered at various deflection angles in a horizontal plane parallel to the table top on which the apparatus sits. The proper definition of anisotropy is the expectation value for cos( ):

It is common to express the definition of anisotropy in an equivalent way:

More to read on anisotropy:

Scattering functions p( )

Scattering functionsECE532 Biomedical Optics


http://omlc.ogi.edu/classroom/ece532/class3/ptheta.html

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The angular dependence of scattering is called the scattering function, p( ) which has units of [sr-1] and describes the probability of a photon scattering into a unit solid angle oriented at an angle relative to the photons original trajectory. Note that the function depends on only on the deflection angle and not on the azimuthal angle . Such azimuthally symmetric scattering is a special case, but is usually adopted when discussing scattering. However, it is possible to consider scattering which does not exhibit azimuthal symmetry. The p( ) has historically been also called the scattering phase function.

The scattering can be described in two ways:

Plotting p( ) indicates how photons will scatter as a function of in a single plane of observation (source-scatterer-observer plane). This pattern is similar to the type of goniometric scattering experiments commonly conducted.

Plotting p( )2 sin indicates how photons will scatter as a function of the deflection angle regardless of the azimuthal angle , in other words integrating over all possible in an azimuthal ring of width d and perimeter 2 sin at some given . The p( )2 sin goes to zero at 0° because the azimuthal ring becomes vanishingly small at 0°. This plot is related to the total energy scattered at a given deflection angle and hence is more pertinent to the value of anisotropy.

Figure depicts a typical forward-directed scattering pattern p( )corresponding to an experimental goniometric measurement in a single source-scatterer-

observer plane,and p( )2 sin which integrates over all possible azimuthal angles .

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Isotropic scattering function




An isotropic scattering function would scatter light with equal efficiency into all possible directions. Such a scattering function would have the form:

Henyey-Greenstein scattering function




Henyey and Greenstein (1941) devised an expression which mimics the angular dependence of light scattering by small particles, which they used to describe scattering of light by interstellar dust clouds. The Henyey-Greenstein scattering function has proven to be useful in approximating the angular scattering dependence of single scattering events in biological tissues.

The Henyey-Greenstein function allows the anisotropy factor g to specify p( ) such that calculation of the expectation value for cos( ) returns exactly the same value g. In other words, Henyey and Greenstein devised a useful identity function. The Henyey-Greenstein function is:

It is common practice to express the Henyey-Greenstein function as the function p(cos);

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A series of Henyey-Greenstein functions are shown in the following figure. The forward direction along the original photon trajectory is 0°. Scattering in the backward direction is 180°. The curve for g = 0 has a constant value of 1/4 .

Reduced scattering coefficient

µs' = µs(1 - g) [cm-1]




The reduced scattering coefficient is a lumped property incorporating the scattering coefficient µs and the anisotropy g:

µs' = µs(1 - g) [cm-1]

The purpose of µs' is to describe the diffusion of photons in a random walk of step size of 1/µs' [cm] where each step involves isotropic scattering. Such a description is equivalent to description of photon movement using many small steps 1/µs that each involve only a partial deflection angle if there are many scattering events before an absorption event, i.e., µa << µs'. This situation of scattering-dominated light transport is called the diffusion regime and µs'is useful in the diffusion regime which is commonly encountered when treating how visible and near-infrared light propagates through biological tissues.

The following figure shows the equivalence of taking 10 smaller steps of "mean free path" mfp = 1/µs with anisotropic deflection angles and one big step with a "reduced mean free path" mfp' = 1/µs'.

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The following figure shows how many such big steps involving isotropic scattering are equivalent to many small anisotropic steps:

Mie theory model for tissue optical properties




Mie theory describes the scattering of light by particles. "Particles" here means an aggregation of material that constitutes a region with refractive index (np) that differs from the refractive index of its surroundings (nmed). The dipole reradiation pattern from oscillating electrons in the molecules of such particles superimpose to yield a strong net

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source of scattered radiation. Also, the reradiation patterns from all the dipoles do not cancel in all but the forward direction of the incident light as is true for homogneous medium, but rather interfere both constructively and destructively in a radiation pattern. Hence, particles "scatter" light in various directions with varying efficiency.

Gustav Mie in 1908 published a solution to the problem of light scattering by homogeneous spherical particles of any size. Mie's classical solution is described in terms of two parameters, nr and x:

the magnitude of refractive index mismatch between particle and medium expressed as the ratio of the n for particle and medium,

nr = np/nmed

the size of the surface of refractive index mismatch which is the "antenna" for reradiation of electromagnetic energy, expressed as a size parameter (x) which is the ratio of the meridional circumference of the sphere (2 a, where radius = a) to the wavelength ( /nmed) of light in the medium,

x = 2 a/( /nmed)

A Mie theory calculation will yield the efficiency of scattering which relates the cross-sectional area of scattering, s [cm2], to the true geometrical cross-sectional area of the particle, A = a2 [cm2]:

s = QsA

Finally, the scattering coefficient is related to the product of scatterer number density, s

[cm-3], and the cross-sectional area of scattering, s [cm2], (see definition of scattering coefficient):

µs = s s

Before using Mie theory to approximate the scattering behavior of biological tissues, let's

briefly examine the Mie calculation illustrate the behavior of Mie scattering with some example calculations and

figures

The math of Mie scatteringECE532 Biomedical Optics


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Consider a source, a spherical scattering particle, and an observer whose three positions define a plane called the scattering plane. Incident light and scattered light can be reduced to their components which are parallel or perpendicular to the scattering plane. As shown in the following figure, the parallel and perpendicular components can be experimentally selected by a linear polarization filter oriented parallel or perpendicular to the scattering plane.

The Scattering matrix describes the relationship between incident and scattered electric field components perpendicular and parallel to the scattering plane as observed in the "far-field" (ref: Bohren and Huffman):

The above expression simplifies in practical experiments:

The exponential term, -exp(-ik(r-z))/ikr, is a transport factor that depends on the distance between scatterer and observer. If one measures scattered light at a contant distance r from the scatterer, eg., as a function of angle or orientation of polarization, then the transport factor becomes a constant.

The total field (Etot) depends on the incident field (Ei), the scattered field (Es) , and the interaction of these fields (Eint). If one observes the scattering from a position which avoids Ei, then both Ei and Eint are zero and only Es is observed.

http://omlc.ogi.edu/classroom/ece532/class3/bh.txt

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For "far-field" observation of Es at a distance r from a particle of diameter d such that kr >> nc

2, k = 2 / , nc = d/ , the scattering elements S3 and S4 equal zero (see Eq. 4.75, Bohren and Huffman).

Practical experiments measure intensity, I = <E E*> = (1/2)a2, where E = a exp(-i ), and a is amplitude and is phase of the electric field.

Hence for practical scattering measurements, the above equation simplifies to the following:

Mie theory yields two sets of descriptors of scattering:

ANGULAR SCATTERING PATTERN OF POLARIZED LIGHT Mie theory calculates the angular dependence of the two elements, S1( ) and S2( ), of the Scattering matrix, from which the scattered intensities of polarized light are computed (see example). The scattering pattern is also used to calculate the anisotropy, g, of scattering by the particle.

Example of angular scattering calculation

Example of anistoropy calculation

EFFICIENCIES OF SCATTERING AND ABSORPTION Mie theory calculates the efficiencies Qs and Qa of scattering and absorption, respectively, such that µa = QaA and µs = QsA, where A is the geometrical cross-sectional area a2 for a sphere of radius a.

Example of Qs and µs calculation

Angular patterns of Mie scatteringECE532 Biomedical Optics



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Consider the scattering pattern from a 0.304-µm-dia. nonabsorbing polystyrene sphere in water irradiated by HeNe laser beam:

np = 1.5721 particle refractive index

nmed = 1.3316 medium refractive index

a = 0.152 µm, particle radius

= 0.6328 µm wavelength in vacuo

As calculated before:

nr = np/nmed = 1.5721/1.3316 = 1.1806 relative refractive index

x = 2 a/( /nmed) = (2)(3.1415)(0.152)/(0.6328/1.3316) = 2.0097 size parameter

Run the Mie theory algorithm:

Mie(nr, x) ---> S1( ), S1( ) as complex numbers

Mie(1.1806, 2.0097) ---> S1.re + jS1.im, S2.re + jS2.im as functions of

To view the results, calculate the intensities of scattering for parallel and perpendicular orientations of polarized source/detector pairs:

Ipar = S2S2* = Re{(S2.re + jS2.im)(S2.re - jS2.im)}

Iper = S1S1* = Re{(S1.re + jS1.im)(S1.re - jS1.im)}

which are shown in the following figures, and can be experimentally measured as described below.

The following figures describe the experimental measurements that illustrate the angular dependence of the scattered intensities Ipar and Iper:

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Iper

Irradiate dilute solution of spheres in water with laser beam polarization oriented perpendicular to the table. Collect scattered light as a function of angle in plane parallel to table. Place linear polarization filter in front of detector perpendicular to the table.

Ipar

Irradiate dilute solution of spheres in water with laser beam polarization oriented parallel to the table. Collect scattered light as a function of angle in plane parallel to table. Place linear polarization filter in front of detector parallel to the table.

Example results: Polar and xy plots of scattering pattern for Ipar and Iper

Polar plot

Click figure to enlarge

I( ) plot


For a randomly polarized light source, the total scattered light intensity is given by the term S11:

http://omlc.ogi.edu/classroom/ece532/class3/gifs/s(th)_polar.gif

http://omlc.ogi.edu/classroom/ece532/class3/gifs/s(th)_xy.gif

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S11 is the first element of the so-called Mueller Matrix, a 4x4 matrix which relates an input vector of Stokes parameters (Ii, Qi, Ui, Vi) describing a complex light source and the output vector (Is, Qs, Us, Vs) describing the nature of the transmitted light. For randomly polarized light, S11 describes the transport of total intensity:

Is = S11Ii

Mie theory calculation of anisotropy (g)




Let's use the scattering pattern calculated previously for a 0.304-µm-dia. nonabsorbing polystyrene sphere in water irradiated by HeNe laser beam to calculate the anisotropy (g) of scattering.

The definition of anisotropy guides the calculation of g using the function S11( ) which describes the scattered intensity for randomly polarized light:

The denominator in the above equation assures proper normalization of p( ):

When numerically evaluating the above expression for g, a large number of angles need to be calculated, typically about 200 angles, in order to achieve a value of g with precision to at least 4 significant digits. For our example 0.304-µm sphere, the calculation of g based on S11( ) yields g = 0.6608.

Note that the definition of g as cited in these notes assumes azimuthal symmetry, hence we have calculated the g for randomly polarized light. Due to the symmetry presented by a spherical particle, the g refers to scattering into all azimuthal angles regardless of the linear polarization of the incident light.

Example: Mie calculation of Qs and µs




Consider the scattering of a HeNe laser beam by 0.304-µm-dia. nonabsorbing polystyrene spheres in water at a concentration of 0.1% volume fraction:

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np = 1.5721 particle refractive index

nmed = 1.3316 medium refractive index

a = 0.152 µm, particle radius

= 0.6328 µm wavelength in vacuo

fv = 0.001 volume fraction of particles in medium

Calculate the following parameters:

refractive index mismatch ratio, size parameter, geometrical cross-sectional area, and number density

nr = np/nmed = 1.5721/1.3316 = 1.1806

x = 2 a/( /nmed) = (2)(3.1415)(0.152)/(0.6328/1.3316) = 2.0097

A = a2 = (3.1415)(0.152)2 = 0.0726 µm2

s = fv/((4/3) a3) = (0.001)/((4/3)(3.1415)(0.152)3) = 0.0608 µm-3

Run the Mie theory algorithm:

Mie(nr, x) ---> Qs

Mie(1.1806, 2.0097) ---> 0.1971

Calculate the scattering coefficient:

s = QsA = (0.1971)(0.0726) = 0.01431 µm2

µs = s s = (0.0608)(0.01431) = 0.0009723 µm-1

µs [cm-1] = (µs [µm-1])(104 [µm/cm]) = 9.723 cm-1

Scattering versus wavelength for 3 particle sizes


©1998Steven L. Jacques, Scott A. PrahlOregon Graduate

Institute

Consider the three optical scattering properties, µs, g, and µs(1 - g), as functions of wavelength for a spherical particle with np = 1.5721 in a medium with nmed = 1.3316 at a concentration of 0.1% volume fraction (fv = 0.001). Assume that np and nmed are constant versus wavelength for this example calculation.

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Let the sphere be one of three sizes: 0.100 µm, 0.300 µm, and 1.00 µm.

The Mie theory calculation yields:

µs


g


µs(1 - g)


Note: The true np and nmed for polysytrene spheres and water are slightly wavelength dependent and would deviate slightly from the above example.

Mie scattering from cellular structures




Soft tissue optics are dominated by the lipid content of the tissues. Such lipid constitutes the cellular membranes, membrane folds, and membranous structures such as the mitochondria (about 0.5 µm). While other objects such as protein aggregates and the nucleus are also sources of scattering, the lipid/water interface of membranes presents a strong refractive index mismatch and so plays a major role is scattering. The following graph summarizes the lipid contents of various tissues:

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Lipid contents of tissues

[ref. 1] Click figure to enlarge

Mie theory provides a simple first approximation to the scattering of soft tissues [ref. 2]. The approximation involves a few assumptions:

Assume the refractive index of the lipid membranes of cells is 1.46, based on the reported 1.43-1.49 range for hydrogenated fats [ref. 3].

Assume the refractive index of the cytosol of cells is 1.35, based on the reported value for cellular cytoplasms [ref. 1].

Assume the lipid content of soft tissue is about 1-10% (fv = 0.02-0.10). Let's choose fv = 0.02 for this example to match the value for several typical soft tissues such as lungs, spleen, prostate, ovary, intestine, liver, arteries, to name a few.

Assume all the lipid is packaged as small spheres of various sizes whose number density maintains a constant volume fraction fv.

Ignore the interference of scattered fields from particles which can alter the apparent scattering properties based on isolated particles.

In the following graphs, a 2% volume fraction of lipid is packaged as spheres of one size, for a series of choices of sphere diameter (radius a), adjusting the number density to maintain a constant volume fraction:

s = fv/((4/3) a3)

Click on figure to enlarge

µs

g

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µs(1 - g)

This picture includes the experimental values of µs(1 - g) for dog prostate, illustrating that the prostate is mimiced by a 2% volume fraction of lipid spheres in the 0.200-0.600 µm diameter range. Other soft tissues vary only a little from this general pattern for prostate tissue.

In summary

The wavelength dependence of light scattering in soft tissues such as the prostate suggests that cellular structures equivalent to spheres with diameters in the range of 0.200-0.600 µm contribute most of the scattering. The magnitude of the scattering suggests that a volume fraction of 0.02 or 2%, which is roughly the reported lipid content of prostate and many other soft tissues, yields a magnitude for µs(1 - g) which matches the magnitude of µs(1 - g) for prostate. Deviations from our assumed values for np based on lipid and nmed based on cytosol and from our assumed value for fv will affect the magnitude of µs(1 - g). But in general,soft tissues with higher (or lower) lipid content will show increased (or decreased) scattering, while the wavelength dependence of µs(1 - g) should not change greatly.

Mie scattering from collagen fibers




Collagen fibers are strongly scattering. For example, the dermis of skin or the sclera of the eye are tissues with high collagen fiber content. Mie theory can be used to approximate the scattering properties of collagen on two levels:

on the macroscopic level of collagen fiber bundles. on the ultrastructural level of periodic striations in collagen fibrils.

macroscopic level of collagen fiber bundles

Collagen fibers vary from 0.1 µm-dia. fibrils to 8 µm-dia. fiber bundles. A study reported the distribution and concentration of collagen fiber bundle diameters in 9 post-mortem neonatal skin specimens [ref.4, Saidi et al. 1995].


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The fiber diameter (d) was 2.80 ± 0.08 µm (n = 9), i.e., (mean ± SD for n specimens), with an intraspecimen variation SD/mean = 0.43 ± 0.05.

The fiber concentration was s = 3 x106 ± 0.5 x106 cm-3. The volume fraction was fv = d2(1 cm) s/4 = 0.21 ± 0.10 (n = 9).

Mie theory can approximate collagen fiber scattering in dermis using the following assumptions:

Assume nmed of the dermal ground substance is 1.350, based on a reported value for the n of corneal ground substance [ref.5].

Assume the np of the collagen fiber bundles is 1.389, based on a mean dermal water content of 65.3%, W = 0.653, (see table 9.1 of ref.1): approximately, n = 1.50 - (1.50 - 1.33)W = 1.389.

Assume one can use the cylindrical Mie theory calculation of Bohren and Huffman [appendix C in ref.6], which assumes that the incident light is oriented perpendicular to the long axis of the fiber cylinders.

Assume that each fiber cylinder scatters light independently, ignoring any interference effects from closely spaced fibers.

The above assumptions allow calculation of µs(1 - g) versus wavelength for dermal collagen fiber bundles. (see "Mie" in figure below).

ultrastructural level of periodic striations in collagen fibrils

The ultrastructure of collagen fibrils presents periodic striations as was shown in the figures in our introduction to scattering. Fibrils are composed of entwined tropocollagen molecules, presenting a banded pattern of periodic striations (70-nm spacing) due to the staggered alignment of the tropocollagen molecules which each have an electron-dense head group that appears dark in the electron micrograph. The periodic fluctuations in refractive index on this ultrastructureal level appear to contribute a Mie scattering component that dominates the visible and ultraviolet wavelength ranges. Such scattering from very small structures is called the Rayleigh limit of Mie scattering, or simply "Rayleigh" scattering.

The following figure compares Mie theory for various size spheres withthe "Rayleigh" component of skin scattering seen experimentally. The "Rayleigh" behavior of skin scattering is mimiced by 50-nm-dia. spheres, np = 1.5, nmed = 1.35, at the volume fraction of collagen in dermis, fv = 0.21. This assignment of the "Rayleigh" scattering to the collagen ultrastructure should be regarded as only a working hypothesis, but there is no other material in large quantity in the dermis to offer a strong source of scattering.


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Combining the Mie and Rayleigh contributions for skin

The combination of the "Mie" and "Rayleigh" contributions to scattering are shown in the following graph, along with measured skin data:

Click on figure to enlarge.

"Mie" contribution is Mie scattering from2.8-µm-dia. cylindrical collagen fiber bundles,np = 1.46, nmed = 1.35, fv = 0.21.

"Rayleigh" contribution is Mie scattering from50-nm-dia. spheres mimicing the ultrastructure associated with the periodic striations of collagen fibrils,np = 1.50, nmed = 1.35, fv = 0.21.

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Diffusion Theory: Introduction

Diffusion theory is the modeling of photon transport due to photon movement down concentration gradients. Diffusion theory is appropriate in medium dominated by scattering rather than absorption so that each photon undergoes many scattering events before being terminated by an absorption event. The photon has a relatively long residence time which allows the photon to engage in a random walk within the medium.

Diffusion theory = Photonsfallingdownagradientof photon concentration.

The instantaneous fluence rate, F(r,t) [J s-1 cm-2] or [W cm-2], is proportional to the concentration of optical energy, C(r,t) [J cm-3] and the speed of light, c [cm/s], in the medium:

F(r,t) = cC(r,t)

The units of energy concentration [J cm-3] times the units of velocity [cm/s] yield the units of fluence rate [J s-1 cm-2]. This relationship between F and C allows us to discuss the time-resolved spatial distribution of photon concentration, C(r,t), in a light-scattering medium using the same math of diffusion which applies to many things such as solutes in a solution or heat in a material.




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Fick's 1st law of diffusion

Diffusion occurs in response to a concentration gradient expressed as the change in concentration due to a change in position, . The local rule for movement or flux J is given by Fick's 1st law of diffusion:

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in which the flux J [cm-2 s-1] is proportional to the diffusivity [cm2/s] and the negative gradient of concentration, [cm-3 cm-1] or [cm-4]. The negative sign indicates that J is positive when movement is down the gradient, i.e., the negative sign cancels the negative gradient along the direction of positive flux.

The flux J is driven by the negative gradient in the direction of increasing x.

Examplearbitrary units

Consider a local concentration of 106 units per cm3 which drops by 10% over a distance of 1 cm. Then the gradient is -105 [units cm-4]. Assume the diffusivity is 103 [cm2/s]. Then the flux J equals:

J = -(103 [cm2/s])(-105 [units cm-4]) = 108 units cm-2 s-1

For light, the diffusivity is proportional to the diffusion length D [cm] and the speed of light c:

= cD

where D = 1/(3 µs(1-g)). The units of velocity [cm/s] times the units of length [cm] yield the units of diffusivity [cm2/s]. The following example describes the local diffusion of red light in milk.

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Exampleoptical energy

Consider a local fluence rate F of 1 W/cm2 in milk (n = 1.33, µs(1-g) = 10 cm-1).

The local concentration of optical energy is C = F/c = (1 W/cm2)/(2.2x1010 cm/s) = 4.4x10-11 J cm-3.

Assume the concentration drops by 10% over a distance of 3 mm. Then the gradient is -(0.10)(4.4x10-11 cm-3)/(0.3 cm) = -1.5x10-11 J cm-4.

The diffusivity equals cD where D = 1/(3(10 cm-1)) = 0.033 cm. Therefore = (2.2x1010 cm/s)(0.033 cm) = 7.5x1010 cm2/s.

Then the flux J along the direction x equals - = -(7.5x1010 cm2/s)(-1.5x10-11 J cm-4) = 0.11 J cm-2 s-1 = 0.11 W cm-2.

For optical diffusion, Fick's 1st law is expressed as the energy flux J [W cm-2] proportional to the diffusion constant D [cm] and the negative fluence gradient dF/dx:

which was obtained by substituting cD for and substituting F/c for C. The factors c and 1/c cancel to yield the above equation.



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Chapter 5CourseHome

Fick's 2nd law of diffusion

Consider diffusion at the front and rear surfaces of an incremental planar volume. Fick's 2nd law of diffusion describes the rate of accumulation (or depletion) of concentration within the volume as proportional to the local curvature of the concentration gradient. The local rule for accumulation is given by Fick's 2nd law of diffusion:

in which the accumulation, dC/dt [cm-3 s-1], is proportional to the diffusivity [cm2/s] and the 2nd derivative (or curvature) of the concentration, [cm-3 cm-2] or [cm-5]. The accumulation is positive when the curvature is positive, i.e., when the concentration gradient is more negative on the front end of the planar volume and less negative on the rear end so that more flux is driven into the volume at the front end than is driven out of the volume at the rear end.

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Incremental planar volume accumulates concentration because the front gradient at x1

drives more flux J1 into the volume than the flux J2 driven out of the volume by the rear gradient at x2.

The differential equation for optical diffusion is simply Fick's 2nd law with the substitution of the product cD for the diffusivity and substitution of F/c for concentration C, although the 1/c factors introduced on both sides of the equation cancel:




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Connecting optical transport theory and Fick's 1st law of diffusion.

The early work on neutron scattering theory in nuclear reactors first developed the connection between transport theory and Fick's 1st law of diffusion. The resulting statement of diffusion is also applied to optical transport. Consider the following problem:

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Light scattered at position r passes through a small aperture with area A contributing to the flux J+.

given a homogeneous isotropically scattering medium with scattering coefficient µs

given a small aperture with area A at the origin (r = 0) given a fluence rate F(0) at the origin assume that the fluence rate F(r) at a position r near to the aperture is

approximated by

calculate the net flux through the small aperture with area A due to scattering from all the surrounding volume:

o the scattered flux at r is µsF(r) = µs( F(0) + r(dF(r)/drat r=0) ) where r is the distance from r to the origin.

o the fraction of scattered flux from r that survives a pathlength r without being scattered or absorbed is exp(-µtr), where µt = µs + µa. Consider the case of negligible absorption compared to scattering so µt = µs.

o the fraction of surviving scattered flux from r that passes through A is Acos /(4 r2).

o integrate the flux from r over all possible and r for the hemisphere above the aperture:

o a similar integration for the scattering from the hemisphere below the aperture yields a positive value J- for flux passing through A in the other direction opposite to J+.

o the net flux J = J+ - J- The final expression for the net flux J has the form of Fick's 1st law of diffusion

if the diffusion constant D has the following value:

, where

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For photon transport, the term µs' = µs(1 - g) is substituted the above µs which described isotropic scattering used in the above treatment.

In summary, the connection between transport theory and Fick's 1st law of diffusion is based on the linearization of F(r) around the value F(0) and on the approximation of D by the value 1/(3 µs').




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Limits of diffusion theory

There are limits to the applicability of diffusion theory.

It is important to remember that diffusion theory simply attempts to mimic Fick's 1st law of diffusion by a proper choice of D to relate J and -dF/dr. The approximation of F(r) as simply a linear perturbation of the value F(0) neglects higher order terms that depend on d2F/dr2. Sometimes the gradient has significant curvature within the spherical regime from which exp(-µs' r) allows photons to pass through the aperture. In such regions, the linear approximation to F(r) is inadequate and diffusion theory is inaccurate. Two cases of curvature deserve emphasis:

Near a source

The gradients are very steep near a point source or collection of point sources with both the exponential term exp(-r/(4cDt)) and the term 1/r causing significant curvature in the gradients. Distant from sources, gradients become gradual and diffusion theory is more accurate.

Absorption

Strong absorption prevents photons from engaging in an extended random walk. The approximation µt = µs' becomes inadequate. In common use is the approximation:

However, we have conducted Monte Carlo simulations of light diffusion from an isotropic point source to yield the spatial distribution for F(r) which suggests a different value for D:




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If µa << 3µs', the effect of absorption can be neglected.




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Time-resolved Diffusion Theory

The solution to Fick's 2nd Law of diffusion for the case of a point source of energy, Uo [units], deposited at zero time (t = 0) at the origin (r = 0) is C(r,t) [units cm-3]:

where r [cm] is the distance from the source to the point of observation, t [s] is the time of observation, and [cm2/s] is the diffusivity. In the denominator, the product t has units of [cm2] which is taken to the 3/2 power to yield units of [cm3]. The exponential in the numerator is dimensionless. The units of Uo can be the units of any extensive variable which is able to diffuse throughout a volume. Hence, C(r,t) has the proper dimensions for concentration, [units cm-3]. The above expression for C(r,t) is a solution to Fick's 2nd law of diffusion and describes spherically symmetric diffusion from a point impulse source in a homogeneous medium with no boundaries.

For the case of optical diffusion, let F = cC and = cD. Let the point source be an impulse of energy Uo [J]. The expression for fluence rate F(r,t) [W cm-2] becomes:

The above equation is very useful and the student of tissue optics should know this equation very well.




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Chapter 5CourseHome

Time-resolved diffusion Theory:

ExamplesThe following figures illustrate the time-resolved transport of light from an impulse isotropic point source of energy within a homogeneous unbounded medium with absorption and scattering properties.

The time-resolved fluence rate F(r,t) is plotted versus r at three timepoints: t = 5, 100, and 1000 ps.

medium: µa = 1.0 cm-1, µs(1 - g) = 10.0 cm-

1, nt = 1.33impulse energy: Uo = 1 J


The same data plotted as a semi-log plot.


The C program that generated the data in the above figures is listed here.




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Steady-state Diffusion Theory

The time-resolved fluence rate F(r,t) [W/cm2] in response to an impulse of energy Uo [J] and the steady-state fluence rate Fss(r) [W/cm2] in response to an isotropic point source of continuous power Po [W] are summarized:

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where the transport factors T(r,t) [cm-2 s-1] and Tss(r) [cm-2] have been introduced to more carefully distinguish the source, the transport, and the fluence rate.

Note on notation: In this class, we use Fss(r) rather than F(r) to especially emphasize the steady-state fluence rate from the time-resolved fluence rate. However, F(r) should be used outside this class.

The above expression for Tss(r) can be obtained by integrating T(r,t)exp(-µact) over all time to yield the total accumulated amount of photon transport to each position r. The factor exp(-µact) accounts for photon absorption (recall that ct = pathlength so this expression is simply Beer's law for photon survival) and causes photon concentration to approach zero as time goes to infinity. The expression for Tss(r) is derived:

Note that the final expression above has made the substitutions:

which removes the diffusion length D and introduces the optical penetration depth which is the incremental distance from the source that causes Fss(r) to decrease to 1/e its initial value. The penetration depth is a parameter which is very easily understood in experimental measurements and consequently has more intuitive value to some people than D which is important from the perspective of the local step size of the diffusion process. In this class we will often use the following expression for Fss(r) when we prefer to emphasize the roles of µa and :




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Steady-state diffusion Theory:

Examples

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The following figures illustrate the transport of light from a steady-state isotropic point source of power within a homogeneous unbounded medium with absorption and scattering properties.

The steady-state fluence rate F(r) is plotted versus r.

medium: µa = 1.0 cm-1, µs(1 - g) = 10.0 cm-

1, nt = 1.33impulse power: Po = 1 W


The same data plotted as a semi-log plot.






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Frequency-domain Diffusion Theory:

Isotropic point source in infinite medium

A light source may be modulated sinusoidally to yield a sinusoidally varying fluence rate distribution at a distant observation point within a medium. Such a modulated concentration will propagate in the medium and is often called a photon density wave. Consider a modulated isotropic point source of light within a homogenous turbid medium with no boundaries.

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The point source S has a steady-state power So [W] which is modulated sinusoidally by a modulation factor Mosin( t), where 0 < Mo < 1:

where = 2 f [radians/s] is the angular frequency of modulation, and the modulation frequency f is in hertz [cycles/s]. The position of observation r is located a distance r from the source. The above equation shows two ways to express S(t), one using a sine function and the other using the equivalent and well-known convention of an exponential with an imaginary exponent.

In response to this modulated source, the modulated fluence rate F(r,t) at r is described:

where Tss(r) is the steady-state transport, k" is the imaginary wavenumber that describes the attenuation of the photon density wave, and k' is the real wavenumber that describes the phase lag of the observed photon density wave.

The expressions for Tss(r), k" and k' are:

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where

The behavior of k' and k" are shown in the following figure which plots k' and k" as functions of /(µac):


The above expressions are equivalent to the expressions published by Schmitt et al. 1992 which in turn are equivalent to the expressions published by Fishkin et al. 1991,1993. Link to references..

At the observation point, the persistence of the source modulation is called the modulation, M, and is often described in the literature as (ACout/DCout)/(ACin/DCin) which equals:

At the observation point, the phase of the signal lags the phase of the source by an angle called the phase, [radians], which equals:

The ratio /(µac) describes the number of radians of modulation cycle that occur in one mean photon lifetime. Only 1/e or 37% of photons survive after a time period of 1/(µac) [s]. is an angle specified by the ratio /(µac), and approaches zero for low modulation frequencies and approaches 90° at the highest modulation frequencies, >> µac.

At very low modulation frequencies, << µac, there is little modulation during the lifetime of a photon. Consequently, photon migration has little impact on the transport of the modulation. The value of k" approaches one so k" approaches 1/ , therefore M approaches unity. The value of k' approaches zero, therefore approaches zero. The observed modulation of Fss(r,t) mimics the source modulation with no loss of modulation and no phase lag. F(r,t) approaches the behavior SoTss(r)(1 + Mosin( t)).

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At high modulation frequencies, as the modulation frequency approaches the value of µac or greater, there is significant modulation during the lifetime of the photon. Consequently, the photon can diffuse during its lifetime and thereby smear the spatial resolution of the modulation. F(r,t) exhibits significant loss of modulation (M < 1) and an increased phase lag ( > 0): F(r,t) = SoTss(r)(1 + MoMsin( t - k'r))




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Frequency-domain diffusion theory:

Examples The following figure illustrates the time-resolved signal F(r,t) in blue and the

source S(t) in red. Note the greatly decreased Fss, the slightly decreased modulation M, and the phase lag at the observation point.

source: f = 400 MHz, So = 1 W, Mo = 1.0medium: µa = 1.0 cm-1, µs(1 - g) = 10.0 cm-1, nt

= 1.33 Click on figure to enlarge

The C program that generated the data in the above figure is listed here.

The following figures illustrates the modulation of the photon density wave as it propagates into the medium.

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The factor Msin(- ) is plotted versus position r. Two values of absorption coefficient µa are used in the calculations, illustrating that lower absorption yields a longer photon lifetime which allows smearing of the spatial resolution of the modulation which attenuates the modulation.Note that very large distances are needed to develop cyclic modulation due to phase lag. Photon density waves do not present multiple cycles in typical experiments. The period of one cycle is rk'/ [s].

source: f = 400 MHz, Mo = 1.0medium: µa = 1.0 or 0.1 cm-1, µs(1 - g) = 10.0 cm-1, nt = 1.33


The fluence rate F(r,t) is plotted versus position r for a specific time (t = 0 [s] or multiples of rk'/ ). The value Tss(r) falls very quickly causing F(r,t) to fall quickly which makes the modulation difficult to see, especially on a logarithmic scale. The modulation is more apparent for the lower absorption.



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