BIOMEDICAL AND ATMOSPHERIC APPLICATIONS OF OPTICAL SPECTROSCOPY IN SCATTERING MEDIA. Johannes Swartling

Transcription

1 BIOMEDICAL AND ATMOSPHERIC APPLICATIONS OF OPTICAL SPECTROSCOPY IN SCATTERING MEDIA Johanne Swartling Doctoral Thei Department of Phyic Lund Intitute of Technology November 00

2 Copyright 00 Johanne Swartling Printed at KFS AB, Lund, Sweden November 00 Lund Report on Atomic Phyc, LRAP-90 ISSN LUTD(TFAF-1050)1-113(00) ISBN

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5 Content Abtract 7 Lit of paper 8 1. Introduction 11. Formulation of the problem The forward problem light propagation model Electromagnetic wave theory Model for ingle cattering baed on electromagnetic wave theory Model for multiple cattering baed on electromagnetic wave theory Tranport theory of radiative tranfer Radiometric quantitie Tranport propertie 3..3 Scattering phae function Reciprocity Solving the tranport equation Polynomial approximation Dicretization method; Adding-Doubling method; Dicrete ordinate Expanion method; The diffuion approximation; The P N -approximation Probabilitic method; Photon migration; Path integral The Monte Carlo method Variation on Monte Carlo imulation Time-reolved and frequency-reolved calculation Fluorecence and inelatic cattering Photon hitting denity and photon meaurement denity function Dicuion olving the forward problem Relationhip between wave theory and tranport theory The invere problem Two-parameter method Spatially reolved diffue reflectance Time-reolved diffue meaurement Three-parameter technique; The integrating phere method Layered media and imple embedded inhomogeneitie Polynomial regreion 64 5

6 4.5 Optical tomography Practical apect and application Tiue optical propertie Scattering propertie of tiue Aborption propertie of tiue - chromophore Water Hemoglobin and myoglobin Lipid Melanin Mitochondrial chromophore cytochrome Dicuion aborption propertie of tiue Optical propertie of blood Tiue phantom Water-baed phantom Rein phantom Refractive index Intrumentation Cw meaurement intrument Frequency-reolved intrument Time-reolved intrument Optical tomography intrument Optical mammography a diagnotic application Atmopheric optic remote ening of trace gae 9 Acknowledgement 95 Summary of paper 96 Reference 99 6

7 Abtract Spectrocopic analyi of cattering media i difficult becaue the effective path length of the light i non-trivial to predict when photon are cattered many time. The main area of reearch for uch condition i biological tiue, which catter light becaue of variation of the refractive index on the cellular level. In order to analyze tiue to diagnoe dieae, or predict doe during, for example, laer treatment, it i neceary to be able to model light propagation in the tiue, a well a quantify the cattering and aborption propertie. Problem of thi type occur in many other area a well, for example in material cience, and atmopheric and ocean-water optic. Thi thei deal with light propagation model in cattering media, primarily baed on radiative tranport theory. Special attention ha been directed to the Monte Carlo model to olve the Boltzmann radiative tranport equation, and to develop fater and more efficient computer method. A Monte Carlo model wa applied to olve a pectrocopic problem in monitoring the emiion of gae in moke plume. An important theme in the thei deal with meaurement of the optical propertie, with emphai on biomedical application. Several different meaurement technique baed on a wide range of intrument have been developed or improved upon, and the trength and weaknee of thee method have been evaluated. The meaurement technique have been applied to analyze the cattering and aborption propertie of ome biological tiue. Much devotion ha been directed to optical characterization of blood, which i an important tiue from a health-care perpective. At preent, the complex cattering propertie of blood prevent detailed optical analyi of whole blood. The work preented here i alo aimed at acquiring a better undertanding of the fundamental cattering procee at a cellular level. 7

8 Lit of paper Thi thei i baed on the following paper: Paper I. Paper II. Paper III. Paper IV. Paper V. Paper VI. Paper VII. J. Swartling, A. Pifferi, A. M. K. Enejder, and S. Anderon- Engel, Accelerated Monte Carlo model to imulate fluorecence pectra from layered tiue, Journal of the Optical Society of America A, in pre (00). J. Swartling, J. S. Dam, and S. Anderon-Engel, Comparion of patially and temporally reolved diffue reflectance meaurement ytem for determination of biomedical optical propertie, ubmitted to Applied Optic (00). J. Swartling, A. Pifferi, E. Giambattitelli, E. Chikoidze, A. Torricelli, P. Taroni, M. Anderon, A. Nilon, and S. Anderon- Engel, Meaurement of aborption and cattering propertie uing time-reolved diffue pectrocopy Intrument characterization and impact of heterogeneity in breat tiue, manucript (00). J. Swartling, S. Pålon, P. Platonov, S. B. Olon, and S. Anderon-Engel, Change in tiue optical propertie due to radiofrequency ablation of myocardium, ubmitted to Medical & Biological Engineering & Computing (00). A. M. K. Enejder, J. Swartling, P. Aruna, and S. Anderon- Engel, Influence of cell hape and aggregate formation on the optical propertie of flowing whole blood, Applied Optic, returned after minor reviion (00). J. Swartling, A. M. K. Enejder, P. Aruna, and S. Anderon- Engel, Polarization-dependent cattering propertie of flowing whole blood, manucript for Applied Optic (00). P. Weibring, J. Swartling, H. Edner, S. Svanberg, T. Caltabiano, D. Condarelli, G. Cecchi, and L. Pantani, Optical monitoring of volanic ulphur dioxide emiion Comparion between four different remote-ening pectrocopic technique, Optic and Laer in Engineering 37, (00). 8

9 Additional material ha been preented in: 1. S. Anderon-Engel, A. M. K. Enejder, J. Swartling, and A. Pifferi, "Accelerated Monte Carlo model to imulate fluorecence of layered tiue," Photon Migration, Diffue Spectrocopy, and Optical Coherence Tomography: Imaging and Functional Aement, S. Anderon-Engel, J.G. Fujimoto, Ed. Proceeding of SPIE Vol. 4160, (000).. J. Swartling, P. Aruna, A. M. K. Enejder, and S. Anderon-Engel, "Optical propertie of flowing bovine blood in vitro," Optical Technique and Intrumentation for the Meaurement of Blood Compoition, Structure and Dynamic In vitro and In vivo. CLEO/Europe 000, Conference Diget p. 354 (000). 3. J. Swartling, C. af Klinteberg, J. S. Dam, and S. Anderon-Engel, "Comparion of three ytem for determination of optical propertie of tiue at 785 nm," European Conference on Biomedical Optic (001). 4. J. Swartling and S. Anderon-Engel, "Optical mammography - a new method for breat cancer detection uing ultra-hort laer pule," DOPS- NYT 4, 19-1 (001). 5. J. Swartling, S. Anderon-Engel, A. M. K. Enejder, and A. Pifferi, "Accelerated revere-path Monte Carlo model to imulate fluorecence in layered tiue," in OSA Biomedical Topical Meeting, OSA Technical Diget, (00). 6. J. Swartling, S. Pålon, and S. Anderon-Engel, "Analyi of the pectral hape of the optical propertie of heart tiue in connection with myocardial RF ablation therapy in the viible and NIR region," in OSA Biomedical Topical Meeting, OSA Technical Diget, (00). 7. M. Ozolinh, I. Laci, R. Paegli, A. Sternberg, S. Svanberg, S. Anderon- Engel, and J. Swartling, "Electrooptic PLZT ceramic device for viion cience application," Ferroelectric 73, (00). 8. M. Soto Thompon, J. Swartling, S. Anderon-Engel, S. Pålon, X. Zhao, Doimetry and fluence rate calculation for fiber-guided intertitial photodynamic therapy: tiue phantom meaurement and theoretical modeling, BiOS 003, San Joe (Accepted). 9

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11 1. Introduction The concept of pectrocopic analyi of material i of profound importance in cience and technology. In traditional pectrocopy, the preence of a ubtance can be detected and quantified by mean of it pectral ignature wavelength band in which light i aborbed (or emitted), defined by the electronic energy level of the atom and molecule that contitute the ubtance. Meaurement of thi kind are performed routinely in thouand every day, to the benefit of the medical ervice, to indutry and a a tool in baic reearch to promote the advancement of our undertanding of nature. The conventional pectrocopic meaurement require that the material i optically clear. A imple definition of a clear material i that the refractive index i contant on patial cale ranging from microcopic, in the order of the wavelength of the light, up to macrocopic. Any patial variation in the refractive index within thi range will catter light in a beam into new direction. To obtain quantitative information from pectrocopy, it i neceary to know the path-length of the light beam through the medium. If the cattering of light i evere, the path-length no longer repreent the hortet ditance from the light ource to the pectrometer through the medium, but a longer one, which i not trivial to predict. Light catter to ome extent in all media, but in many cae the effect i o mall that it may be neglected. In an intermediate regime, the cattering may be ignificant, but till mall enough o that the aumption of a clear medium can be ued with uitable correction. One of the main objective of thi thei i to deal with the prediction of the light path-length through media where the cattering i o trong that uch correction are no longer valid. There i no clear delineation where the weakly cattering regime top and the trongly cattering regime tart it depend on the problem. Often, one talk about multiple cattering. If light i regarded a photon, multiple cattering occur when there i a large probability that any given photon in a beam will catter more than once. Then it i evident that a trongly cattering medium i characterized by two thing: the probability of cattering, and the dimenion of the medium. For example, a piece of paper ha a very high probability of cattering, and catter light trongly even though it phyical dimenion are mall. On the other hand, the probability of cattering in the atmophere i comparably low, but taken over everal kilometer, the cattering of a light beam may till be ignificant. Another objective of thi thei i to how that the ame model and principle may be applied to very mall geometrie, uch a heet of paper, and very large one, uch a atmopheric meaurement. Example of trongly cattering media that are intereting from a pectrocopic point of view include the already mentioned paper and atmophere, ocean water, a 11

12 large number of olid material around u, and mot biological tiue. Since the invention of the laer in the 1960, and it rapid adoption by the medical community, tiue optic ha been the major field of reearch where trongly cattering media have been tudied. In the tudy of tiue optic, the third main objective of thi thei become clear: the cattering of a material i not only a nuiance in pectrocopic meaurement. By analyzing the cattering propertie, important information of the material may be obtained. Thu, the cattering itelf become an object of analyi rather than jut a parameter to be controlled. Although reearch in tiue optic took off during the 1970, it i of hitorical interet to note that much of the fundamental theory wa developed earlier, in other branche of phyic. A will be decribed in thi thei, light cattering i uually modeled from a tarting point of either of two theorie: wave theory (Maxwell equation) or tranport theory. Much of the theory of cattering from ingle particle uing wave theory wa developed in the early year of the 0 th century. Tranport theory originate from the late 19 th century, but the development wa accelerated with the need to model neutron tranport in nuclear reactor. Much of the relevant literature and computer code ued for light propagation wa inherited by reearch in neutron tranport. In medical application of light and laer, the whole range of important iue of light propagation in cattering media i demontrated. A fundamental categorization i the forward problem and the invere problem. The forward problem i, given that the optical propertie of the medium are known, to predict how light will propagate through the medium. The practical importance of thi in medical application i mainly in therapy, for example to calculate the doe in a laer treatment. The invere problem i, given that light that ha penetrated the tiue i meaured, to deduce what the optical propertie inide the medium are. The practical importance here i mainly in diagnotic, ince both the aborption propertie the traditional pectrocopic ignal and the cattering propertie carry information on the tate of the tiue. The other important apect of the invere problem i to provide input data for calculation of the forward problem. However, a will be een, it i not poible to olve the invere problem without firt olving the forward problem. Thi thei concern model of light propagation in cattering media, with the emphai on tranport theory. Specifically, the Monte Carlo method wa ued extenively in Paper I and VII. Meaurement of the optical propertie i another major part of the thei, and Paper II VI are devoted to thi problem. Mot work ha been done within the framework of potential practical application, primarily in medicine (Paper I VI) but alo in environmental monitoring (Paper VII). 1

13 . Formulation of the problem From a very general perpective, conider Fig..1. A turbid medium, delineated by a boundary, i illuminated from the outide, or by light ource from inide, with light X in (r,,t). The denotation X repreent ome uitable radiometric quantity, r repreent the patial coordinate, i a direction and t time. For implicity the light i aumed monochromatic. The medium ha optical propertie denoted p(r), for the moment diregarding their phyical origin. It i uually aumed that p(r) i quai-contant in time, i.e., any change in the optical propertie occur on a longer time cale than the propagation of light. Light that either propagate through the medium or ha emerged i denoted X prop (r,,t). The Forward Problem The firt tak i to find a way to predict X prop (r,,t), given that we know p(r). Thu, we want to find the tranfer function f: f[x in (r,,t);p(r)] X prop (r,,t) (.1) The Invere Problem The next tak i to find a way to deduce p(r), given that we have meaured X prop (r,,t), or ome part of it. Thi mean finding the invere to the above: f -1 [X in (r,,t);x prop (r,,t)] p(r) (.) Thee problem comprie the fundamental quetion poed in thi thei. In order to olve the forward and the invere problem for a turbid medium, a number of phyical theorie, aumption and approximation are needed. In the next chapter, the forward problem will be dicued, followed by the invere problem in Chapter 4. To conclude, ome practical apect concerning tiue optic, intrumentation iue, and application are dicued in Chapter 5. 13

14 X (r,,t) in p(r) X prop (r,,t) Fig..1 The geometry of light propagation in a turbid medium, in general term. The medium i delineated by a boundary, and it i illuminated by light repreented by X in (r,,t). The light that propagate through the medium, or ha emerged, i denoted X prop (r,,t). The optical propertie are denoted p(r). 14

15 3. The forward problem light propagation model The definition of the optical propertie p(r) in Fig..1 depend on the phyical theory one chooe to decribe the light propagation. A mentioned in the introduction, either of two phyical theorie of light i conidered: wave theory or tranport theory. Wave theory, or electromagnetic wave theory, relie on olution of the Maxwell equation. In thi context, the optical propertie are defined by the complex dielectric contant, ε(r). The variation in Re{ε(r)} decribe the cattering, while Im{ε(r)} repreent the aborption propertie. Only in pecial cae i it poible to olve the wave equation for large macrocopic problem, a will be dicued later. In mot problem, epecially thoe related to tiue optic, it i intractable both to olve the wave equation and to cope with the vat complexity of the variation of ε(r) on a microcopic level. To deal with uch problem, the tranport theory of radiative tranfer i better uited. In tranport theory, light i heuritically regarded a energy propagating according to the rule defined by the tranport equation, a conceptually imple equation of conervation. The optical propertie p(r) are defined by mean of a cattering coefficient, an aborption coefficient and a cattering phae function which relate to the probability of cattering in different direction. In the next ection, ome relevant part of electromagnetic wave theory, tranport theory and their relation will be decribed. Becaue of the vat number of publication on the baic theory of thee ubject already available, the following treatment will focu on the ue of the model rather than full theoretical derivation. 3.1 Electromagnetic wave theory Maxwell equation form the tarting point of the decription of light propagation a electromagnetic wave propagating through a dielectric medium. The field are claically decribed by: B E = (3.1) t D H = + F t (3.) D = ρ (3.3) B = 0 (3.4) 15

16 where E [Vm -1 ] and H [Am -1 ] are electric and magnetic field vector, B [Vm - ] i the magnetic flux denity vector, D [Am - ] i the electric diplacement vector, F [Am - ] i the current denity vector (the conventional notation J i in thi thei reerved for the radiometric quantity radiant flux denity, ee Eq. (3.8) (3.11)), and ρ [Am -3 ] i the volume charge denity. The electric and magnetic field can be related to the diplacement field and flux denity field by contitutive relation, depending on the propertie of the medium. In a non-diperive iotropic medium, which we are intereted in here, the relation are D = εe and B = µh, where ε [AV -1 m -1 ] i the permittivity and µ [VA -1 m -1 ] i the permeability. The current denity and the electric field are alo coupled by F = σe, where σ [AV -1 m -1 ] i the conductivity. The Maxwell equation can be olved directly uing numerical method, which will be dicued below. The computation for large problem are daunting, however, and clever ue of expanion method and approximation can greatly reduce the calculation needed for ome problem. Typically, it i aumed that the medium i nonconductive, and one can then derive the vector wave equation (or Helmholtz equation) 1 : where k = π/λ i the wavenumber (λ i the wavelength). E + k E = 0 (3.5) H + k H = 0 (3.6) Often, one i intereted in prediction of the cattering from ingle particle, both becaue many cattering media in fact conit of enemble of particle, and alo becaue ometime it i poible to aume that a cattering medium may be approximated by cattering particle. Scattering from particle can be decribed in term of diffraction,3 or approximation uch a thoe preented by Rayleigh- Gan-Debye -4, but more general approache are given by Mie theory and T-matrix theory. Mie theory deal with pherical particle, while T-matrix theory i applicable to particle of arbitrary hape, although in practice only particle of pheroidal ymmetry are ueful. The general idea in Mie and T-matrix theory i to expand the field in vector pherical function Model for ingle cattering baed on electromagnetic wave theory Mie theory (or Lorenz-Mie theory) provide a quick and relatively imple way of calculating light cattering,3. The relevant input parameter to a Mie calculation are the ratio of the refractive index inide the particle to that in the urrounding medium, m = n particle /n medium, and the ize parameter x = πa/λ, where a i the 16

17 radiu of the particle. The reult of a Mie calculation i a map of the cattered field from an incident plane wave. Uually, one i intereted in the extinction cro ection C ext [m ], cattering cro ection C ca [m ], and the aborption cro ection C ab [m ] of the particle. Thee can be obtained through integration of the cattered field, and are related a C ext = C ca + C ab. It i alo convenient to define relative extinction, cattering and aborption cro ection, Q ext, Q ca and Q ab (dimenionle). The relative cattering cro ection i defined a Q ca = C ca /πa, and the other analogouly. Another property of interet i the cattering aniotropy factor, g = <coθ>, where θ i the cattering angle. The aniotropy factor i a meaure of how cloe to iotropic the cattering i. For entirely iotropic cattering, uch a Rayleigh cattering, g = 0. In thi context it can alo be noted that Mie theory collape to the claical Rayleigh expreion for cattering when x 0 (cf. Eq. 5.6). The applicability of Mie theory on a problem depend on everal factor. Particle that are pherical by nature are of coure prime candidate. Example of thi kind are liquid aerool uch a water droplet. Particle of irregular hape can alo be modeled uccefully uing Mie theory under certain condition. Several tudie have hown that in an enemble of randomly oriented particle of nonpherical hape, the average cattering can often be repreented by Mie theory of phere of equivalent ize. However, thi i not alway the cae, a ome author have pointed out 5. Mie theory i alo important for validation purpoe. Intrument deigned to meaure the cattering of a medium can be teted on ample with microphere with known ize and refractive index, to erve a a verification againt theory. Thi i dicued in more detail in Sect. 5.. Finally, Mie calculation are ueful to provide quick and approximate reult when only order-of-magnitude number are needed for media that conit of irregular cattering tructure. Mie calculation i not entirely trivial, and the computation are uceptible to round-off error in the numerical routine. New Mie code therefore have to be teted thoroughly. For thi reaon, it i uually bet to try to find an exiting, wellteted program. In thi thei, all Mie calculation were performed uing the program by Bohren and Huffman, BHMIE 3. Mie program are available on the Internet, alo a interactive web cript 6. T-matrix theory preent a more general method to calculate cattering from particle of other hape than pherical 5,7. In principle, any hape i poible, but due to the fact that the field vector expanion i baed on vector pherical function, pheroidal particle are bet uited. The calculation are even more enitive to round-off error than Mie theory, epecially a the ize parameter increae. For practical purpoe, only particle of ome pherical ymmetry are poible becaue of thi. T-matrix calculation were performed to tudy the cattering from red blood cell in Paper V. A modified verion of the code by 17

18 Barber and Hill wa ued 8,9. Single-preciion (16 digit) T-matrix computation are poible for ize parameter x < 5, but with extended preciion (3 digit) ize parameter up to around x = 65 are poible with good accuracy Model for multiple cattering baed on electromagnetic wave theory Enemble of particle are poible to model uing Mie or T-matrix theory, a long a the ditance between the particle are large. The total cattering coefficient can then eaily be calculated, becaue the individual particle are in the cattering farfield with repect to their neighbor. When the interparticle ditance become mall, the particle tart to influence each other in their near-field, and the aumption of ingle, independent catterer break down. In ome cae, aggregate of a mall number of particle are poible to model uing Mie or T- matrix theory uing a uperpoition approach 7,11, but for more complicated geometrie more general method are required. The perhap mot traightforward method of olving a wave problem for an arbitrary geometry i by dicretizing Maxwell equation, the patial coordinate and time. Thi method i called Finite Difference Time Domain (FDTD), and can in principle olve any problem. However, due to the computational requirement, FDTD i limited to rather mall problem. A a rule of thumb, the patial dicretization mut be λ/15 or maller, which mean about 10 6 point for a problem of ize 5λ. For each time tep, one operation i required for every point in pace. More information on FDTD can be found in Ref. 1 and 13. Calculation of light cattering from ingle biological cell uing FDTD ha been demontrated An alternative approach to FDTD i to ue the Finite Element Method (FEM) to olve Maxwell equation. In general, FEM i bet uited to olve partial differential equation on cloed domain, i.e., boundary value problem. FEM require the medium to be repreented by a meh, and one of the principal advantage of the method i the veratility of the meh deign and flexibility of repreenting complicated hape and variation in dielectric contant. Another advantage of FEM i that the matrice are typically pare, o that the numerical machinery that pertain to pare matrix computation can be utilized. A drawback of the method i that pecial care ha to be taken when modeling unbounded domain, to terminate the meh uing the proper boundary condition. Several commercial and free FEM code are available, ranging from very imple D repreentation to advanced package. Example of free code are EMAP 17 and Student QuickField 18, while commercial oftware package include FEMLAB (a Matlab toolbox) 19 and EMFLex 0. 18

19 A lightly le direct approach i preented by the Method of Moment (MoM). In thi method, the problem i reduced to maller domain, where the Maxwell equation are formulated a integral equation 1. An example of a free MoM code i PCB-MoM. The method mentioned o far uffer from being retricted by the computational reource required for problem larger than a few wavelength. Larger problem, up to a few hundred wavelength, can be olved uing the Fat Multipole Method (FMM) 3. FMM utilize an efficient method for numerical convolution of the Green function for the vector wave equation, which lead to a reduction of the numerical complexity. The method doe not inherently involve any approximation, but by utilizing problem-pecific propertie the computation can be made even more efficient. One uch aumption may be that the variation in refractive index in the medium i mall. Thi condition i fulfilled in human blood, which render FMM a poible candidate for modeling the complex cattering propertie of blood (ee alo Sect ; Optical propertie of blood). To olve even larger problem, approximation method can be ued. The approximation method are ometime denoted aymptotic method, which in turn can be categorized into four clae: approximation of partial differential equation and integral, geometrical optic, phyical optic, and other method. A an example from the firt area, the vector wave equation, which i elliptic, may be approximated by a paraxial equation, which render the parabolic equation method 4. Thi method i uitable for large problem with a clear, preferred direction of propagation. A cloely related approach i the Bremmer erie method 5. Geometrical optic i valid when the curvature of the object i large compared with the wavelength, i.e., typically for large object. The ray trajectorie are given by the famou Fermat principle, tating that the path of a ray i alway uch that the optical path length i minimized. Geometrical optic problem can be olved uing ray tracing oftware. Phyical optic depend on integral repreentation of the far field, for catterer that are perfectly conducting. The requirement of large object hold for phyical optic a well. The two method can be combined with other method, uch a MoM, if maller object are involved in the problem. The lat category, other method, include imple optical model uch a ray tracing without a phae front, and Fraunhofer and Frenel diffraction. The reult from the wave equation can alo be ued a a tarting point for a rigorou, analytical derivation of tatitical quantitie relevant for multiple cattering problem. Thi lead to differential or integral equation that, in principle, include all wave effect. However, the olution are complicated and in practice variou approximation are employed. Example of method include 19

20 Twerky theory, the diagram method, and the Dyon and Bethe-Salpeter equation. An overview of analytical theory i given by Ihimaru 6. Twerky theory ha been applied on the problem of light cattering in human blood 7,8 (cf. Sect ; Optical propertie of blood). However, the reult of Twerky theory i equation with parametric dependence, where the parameter cannot be eaily deduced from conideration of the fundamental geometrical and dielectric propertie of the medium. In term of practicality, the theory i therefore more imilar to tranport theory, which i the topic of next ection. 3. Tranport theory of radiative tranfer The radiative tranport equation (RTE) (or Boltzmann equation) can be tated a 1 L( r,, t) = c t = L( r,, t) ( µ a + µ ) L( r,, t) + µ 4π L( r,, t) p(, ' ) d ω( ) + q( r,, t) (3.7) The RTE i an equation of conervation, decribing the change in radiance L in the direction inide a mall volume element dv. Thu, the firt term on the right hand ide decribe the loe over the boundary of dv, the next term the loe due to aborption and cattering, the third term the gain due to cattering from other direction into, and the lat term gain due to any ource q inide dv 9,30. Defining the remaining deignation introduced, tarting from the left, we have the light peed in the medium c [m/], the aborption coefficient µ a [m -1 ], the cattering coefficient µ [m -1 ], and the cattering phae function p(,') [-]. The cattering phae function give the probability of cattering from direction ' into direction. In the integral, dω() denote an infiniteimal olid angle in the direction. Claical, and till eential, reference on tranport theory include the work by Chandraekhar 31, Cae and Zweifel 9, and Ihimaru 6. A recent treatment, oriented toward tiue optic, i given in Ref Radiometric quantitie Before dicuing the RTE further, it i ueful to define ome radiometric quantitie and their relationhip. The radiant flux denity J [W/m ] i defined a the power P tranferred through a urface area A: 0

21 P = J n d A, A (3.8) where n denote the normal vector to the urface element da. The calar quantity inide the integral, i.e., the power per unit area, i called the irradiance E(r,t) [W/m ]: E ( r, t) = J( r, t) n( r) (3.9) The intenity I(r,,t) [W/r] i defined a the power per unit olid angle. The radiance L(r,,t) [W/m r] i defined a the power per unit olid angle and area. The relationhip between J and L i given by J( r, t ) = L( r,, t) d ω( ). 4π (3.10) The hemipherical flux, which i the flux through the area element da in either direction, i a ueful quantity. It i defined a J n + ( r, t) = L( r,, t)( n) d ω( ). π (3.11) If the hemipherical flux i meaured from a urface, it i called the radiant exitance or emittance [W/m ]. In tranport theory, light tranport i often regarded a a tranport of photon, interpreted a claical particle. Although the RTE doe not inherently pecify the nature of the tranported energy a particle, there are everal reaon for thi interpretation. Hitorically, neutron tranport wa the major field where method in tranport theory were developed. The context i thu uited for a particle interpretation. In addition, in the Monte Carlo method, a will be apparent in Sect , the particle repreentation i natural. For thee reaon it i convenient to define a photon ditribution function N(r,,t) [m -3 r -1 ], which i the volume denity of photon per unit olid angle. The relationhip between the radiometric quantity L and the photon denity N i then where h i Planck contant. hc L( r,, t) = N( r,, t), (3.1) λ Another important quantity i the fluence rate φ [W/m ], which decribe the power incident on a volume element per urface area: 1

22 φ( r, t ) = L( r,, t)d ω( ). 4π (3.13) The fluence rate i ueful ince by knowing the aborption in the medium, the aborbed energy W [J/m 3 ] can be calculated a W ( r ) = µ a ( r ) φ ( r, t) dt. (3.14) Thi equation i important, ince it couple the depoited energy doe in a medium, to the radiometric quantity fluence rate. E E 0 µ t E = E exp(-µ d) 0 t d x Fig. 3.1 Illutration of Beer-Lambert law. 3.. Tranport propertie Returning to the dicuion about the RTE, one identifie four medium-dependent parameter: the light peed c determined by the refractive index, the aborption and cattering coefficient µ a and µ, and the cattering phae function p(,'). The coefficient µ a and µ hould be interpreted a the probability of aborption and cattering per unit path length, repectively. Their meaning i clear when conidering the generalized Beer-Lambert law, [ ( µ + d] E = E exp, (3.15) 0 a µ )

23 which decribe the attenuation of a collimated beam (plane wave) of initial irradiance E 0 through a medium of thickne d (ee Fig. 3.1). Within the framework of the particle interpretation, the reciprocal of µ a + µ, 1/(µ a + µ ), i interpreted a the mean free path length between photon interaction with the medium. The quantity µ t µ a + µ i called the total attenuation coefficient Scattering phae function The cattering phae function p(,') decribe the angular probability of cattering from direction ' to. The phae function i ometime written a p(coθ) to emphaize the angular dependency, and although thi i only poible when there i no abolute directional dependency, phyically realitic phae function virtually alway only exhibit relative angular dependency. It i uually aumed that the cattering probability i ymmetric for the azimuthal angle ψ, although thi i not a trict requirement. The phae function i normalized: 1 p(coθ) d(coθ) = 1. (3.16) 1 Fig. 3. Scattered field from pherical particle calculated with Mie theory. In the calculation, m = 1.5, and x = π. The aniotropy factor i g = To exemplify the importance of the phae function, conider the cattering from a microcopic phere, a decribed by Mie theory (ee Fig. 3.). A a general rule, a the diameter increae the cattering get increaingly forward-favored. Lobe are 3

24 viible in certain angle due to interference effect. In a polydipere enemble of particle, the lobe average out and the phae function become more or le mooth. The mot widely ued phae function to approximate thi hape i the Henyey-Greentein phae function 3, which ha the functional form (1 g ) p(coθ) = (1 + g g coθ) 3/, (3.17) where g i called the cattering aniotropy factor or imply g-factor, and i defined a g = <coθ>. The hape of the Henyey-Greentein function i hown in Fig. 3.3 for three value of g. The g-factor can be calculated for any phae function, and i a meaure of how forward-favored the cattering in the medium i. Other phae function have alo been ued in the literature, uch a the Reynold- McCormick phae function (alo called Gegenbauer-kernel phae function) 33. It i alo poible to directly incorporate phae function from Mie or T-matrix computation, which will be dicued more in connection with Monte Carlo imulation (Sect ). g = 0 g = 0.5 g = 0.8 Fig. 3.3 The Henyey-Greentein phae function for three different value of the cattering aniotropy factor g Reciprocity Before going into the variou method of olving the tranport equation, the concept of reciprocity within tranport theory will be dicued. Let u, for now, only recognize the fact that many numerical olution to tranport problem tart with point-like light ource, and the olution evolve during the computation a a point preading proce. Real light ource are patially finite, and it may be neceary to convolve thi Dirac repone with the actual patial hape of the ource. In a large cla of problem, however, the light ource i ditributed over a volume, and the detector i almot point-like and may alo be directional. Thi kind of problem may be computationally very inefficient to model in a traightforward 4

25 way. The reciprocal ituation could then be a much more efficient model, provided that one can how that the two ituation are equivalent. Reciprocity wa ued in both Paper I and VII, and therefore a more detailed derivation of the reciprocity theorem within tranport theory will be preented here. The derivation eentially follow that in Ref. 9. Conider the RTE, Eq. (3.7). The time-dependent RTE can alway formally be reduced to a time-independent equation through a Laplace tranform 9. Therefore, we only have to derive the reciprocity theorem for the time-independent RTE: L( r, ) + ( µ ) L( r, ) = µ L( r, ' ) p(, ' ) d ω( ) a + µ + 4π Q( r, ). (3.18) Let L 1 (r,) be the unique olution to Eq. (3.18) for a given ource Q 1 (r,) and incident ditribution L inc,1 (ρ,) on the urface S of the volume V: L1 ( r, ) + ( µ a + µ ) L1 ( r, ) = µ L1 ( r, ' ) p(, ' ) d ω( ) + Q1 ( r, ), 4π L ρ, ) = L ( ρ, ), n 0. 1 ( inc, 1 < (3.19) ~ A unique olution alway exit if µ a > 0. Let L 1( r, ) be the olution to an RTE identical to (3), except that p(, ') p( ', ), (3.0) i.e., ~ L ( r, ) + ( µ ~ ) L ( r, ) = µ ~ L ( r, ' ) p( ', ) d ω( ) Q ( r, 1 a + µ π ~ L ( ρ, ) = L ( ρ, ), n 0. 1 inc,1 < ), (3.1) Now, if the phae function p i invariant under time reveral, we have p( ', ) = p(, '), (3.) ~ and it i clear that L 1( r, ) = L1 ( r, ) ince they are both unique olution to the ame equation with the ame boundary condition. Furthermore, we can define two ~ olution L (r,) and L ( r, ) in a imilar way. Since we are deriving an expreion ~ for reciprocity, the quantity we are intereted in i L ( r, ). Thi give the equation 5

26 6 ),, ( ) ( ) d ', ) p( ', ( ~ ), ( ~ ) ( ), ( ~ 1 4 r r r r + ω = µ + µ µ + π Q L L L a 0 ),, ( ), ( ~, > ρ = ρ n inc L L. (3.3) Now, multiply Eq. (3.19) by ), ( ~ r L, and integrate over V and : π π π π π ω + ω ω µ = ω + µ µ + ω V V V a V V L Q V L L V L L V L L ) d ( ) d, ( ~ ), ( ) d ( ) d ' ( ) d ', ) p( ', ( ), ( ~ ) d ( ) d, ( ~ ), ( ) ( ) d ( ) d, ( ~ ), ( r r r r r r r r (3.4) Similarly, multiply Eq. (3.3) by L 1 (r,) and integrate over V and, and ubtract from Eq. (3.4). The divergence term can be implified to a urface integral uing Gau theorem, and we obtain: { } { } π π π π ω ω µ + ω = ω V V S dv L L L L V L Q L Q S L L ) ( ')d ( d ) ', ) p(, ( ~ '), ( '), ') p(, ( ), ( ~ )d ( d ), ( ), ( ), ( ~ ), ( )d ( )d, ( ~ ), ( ) ( r r r r r r r r r r n (3.5) The lat term vanihe becaue we can make the variable tranformation '. Since we had aumed that p(,') wa invariant under time reveral, and thu that ), ( ), ( ~ r r L L =, we finally obtain { } π π ω = ω V S V L Q L Q S L L d d ), ( ), ( ), ( ), ( d ) d, ( ), ( ) ( r r r r r r n (3.6) Equation (3.6) expree the reciprocity theorem on integral form. Proceeding to derive the reciprocity theorem in the cae uually encountered in tiue optic, conider the geometry in Fig 3.4. It i clear that Q 1 i an iotropic ource inide the volume V: ) ( 4 ), ( r r r δ π = P Q. (3.7)

27 (a) Forward n (b) Revere n Q n 1 ω n 1 ω n r n r Direction of propagation Direction of propagation r O r Q O Fig. 3.4 Reciprocity ued in tiue optic. The refractive indice outide and inide the medium are denoted n 1 and n, repectively. The normal vector at the urface i denoted n. In (a), the forward cae, we have an iotropic light ource Q 1 at r 1 that give rie to a flux over the boundary at r. The radiance at r i integrated over the olid angle ω, which may be determined by the condition for total reflection, or by the collection angle of a detector at r. In (b), the revere cae, a urface ource Q at r give rie to a fluence rate at r 1. The ource Q emit in the olid angle ω. where P 1 i the power emitted by the ource. For the reciprocal cae, the definition of the light ource i le obviou. Apparently, we could define an incident light on the boundary L inc,1 (ρ,) and let Q be zero. However, it turn out that it i alway poible to replace an incident light ditribution with an equivalent urface ource 9. Thi mean that the left-hand ide of Eq. (3.6) vanihe, and we can define a urface ource Q a P ( n) δ( r r ) r if i inide ω Q ( r, ) = F ω (3.8) 0 if i not inide ω where P i the emitted power, r F i a factor that account for Frenel reflection at the interface, and the olid angle ω i defined by the critical angle for total reflection at the boundary (or the collection angle of a detector at r ). In cae the refractive indice are equal, ω = π. Hence, we get: P1 4π 4π P L ( r1, ) d ω( ) = ω ω L ( r 1, )( n) r F d ω( ). (3.9) 7

28 The integral on the left-hand ide i exactly the fluence rate at r 1 due to the urface ource Q (r,), while the integral on the right-hand ide i exactly the radiant flux denity at the urface at r due to the iotropic ource Q 1 (r,). In practice, we are intereted in the cae when thee two quantitie are equal, and we get P = ω 1 4π P. (3.30) Thu, to get the ame reult from two reciprocal computation, the power of the two reciprocal ource hould be caled according to Eq. (3.30). A more detailed derivation of Eq. (3.7) (3.30) i given in Paper I. A we have een, the reciprocity theorem i valid under the aumption that the phae function i invariant under time-reveral, p( ', ) = p(, '). (3.31) A natural quetion i whether there are any phyically relevant phae function that do not exhibit thi kind of invariance. Starting with the Henyey-Greentein phae function, Eq. (3.17), we ee that there i no dependence on the direction and thu we are free to make the variable ubtitution in Eq. (3.31) without violating the equality. The ame i true for any phae function computed from Mie theory, which i clear becaue of the ymmetry of pherical particle. For any normal cattering condition it eem that we can aume that the time-reveral invariance of the phae function hold Solving the tranport equation A range of different technique to olve the RTE are available, each with it advantage and drawback. Firt, we note that no analytical olution to the RTE are available in 3D, for any geometry other than uch that can be repreented in one or two dimenion. Full olution of the RTE are only poible uing numerical method, e.g. by dicretization of the equation. The mot widely ued dicretization method i the dicrete ordinate method, which will be decribed in Sect The other option i the ue of Monte Carlo imulation, a method that ha been widely adopted by the tiue optic community. Intead of attempting a full olution, variou method baed on implification or approximation are available. Sometime, the dimenionality of the problem can be reduced. For a few, pecial, but important geometrie, polynomial approximation have been developed. Perhap the mot important approximation i the diffuion 8

29 approximation, which i baed on the firt term in a pherical harmonic expanion. In the next few ection, emphai will be turned to the Monte Carlo imulation method, but mot of the other important method for olving the tranport equation will be reviewed or at leat be given reference to. A before, the treatment focue on the practical apect of the method rather than derivation, which can be found in the reference Polynomial approximation Polynomial approximation have no phyical meaning and are not olution to the RTE per e, but they may be ueful tool for quick calculation. The idea i to find a polynomial expreion decribing the optic of the medium uing one parameter. A ueful example i the total reflectance from a emi-infinite medium, illuminated with diffue light. Thi ha been found to follow 34 where (1 )( ) R =, (3.3) a = 1 ag (3.33) and a i the albedo, a = µ /(µ a +µ ). The error of prediction ha been hown to be le than for any combination of µ, µ a and g. More on polynomial approximation can be found in Ref. 34. Approximation for collimated incident light, alo for index mimatch between the emi-infinite media, can be found in Ref Dicretization method; Adding-Doubling method; Dicrete ordinate A already dicued in connection with the vector wave equation, the mot traightforward way of olving complex equation i by direct dicretization and ubequent numerical computation. A firt tep in dicretization of the RTE i to dicretize the radiance in angular component, 1,,... N. The equation can then be written a 9

30 L( r, i i ) + µ t L( r, ) = µ w j L( r, ) p(, ) + i N j= 1 j i j Q( r, i ), (3.34) where w j are weighting factor ued in the quadrature. Thi general approach i called the dicrete ordinate method or the N-flux method. The implet way of dealing with thi equation i to include only one angular component, the forward direction. In thi context the radiance i not a ueful quantity ince it i defined by mean of olid angle. Intead, one mut ue the irradiance. The RTE i then reduced to de( x) dx = µ t E( x), (3.35) which ha the olution recognized a Beer-Lambert law. E x) = E( x = 0) exp( xµ ) (3.36) ( t Increaing complexity lightly, we include two angular component, the forward and the revere direction. Thi i the -flux, or one-dimenional, tranport theory. The one-dimenional tranport equation i a et of coupled differential equation: de+ ( x) = ( µ a1 + σ) E+ ( x) + σe ( x) (3.37) dx de ( x) = ( µ a 1 + σ) E ( x) + σe+ ( x). (3.38) dx Here, E + (x) propagate in the poitive x direction, and E - (x) in the negative. µ a1 i the one-dimenional aborption coefficient, and σ = µ 1 p( x,x) = µ 1 p(x, x), where µ 1 i the one-dimenional cattering coefficient. A full derivation of Eq. (3.37) and (3.38) can be found in Ref. 30. A hitorically important verion of onedimenional tranport theory i given by the Kubelka-Munk theory 36, which aume diffue light flux. If the cattering dominate over aborption, one can how that the one-dimenional propertie are related to their three-dimenional counterpart by µ a1 µ a =, (3.39) 30

31 a + µ ( 1 g ) = ( µ 1 + σ). (3.40) 3 µ a Kubelka-Munk theory wa ued extenively in the early day of tiue optic, and till find application. A modern example where Kubelka-Munk theory i ued i for rendering kin and other cattering urface in computer graphic, uch a video game and pecial effect in motion picture 37. Publication of later date tetify that the method may till be ueful for ome application in tiue optic 38. The olution to Eq. (3.37) and (3.38) depend on the boundary condition. Solution for variou geometrie can be found in Ref. 30 and 39. The next tep in complexity for olution of the RTE i preented by the addingdoubling method, which aume cylindrical ymmetry. The radiance i dicretized in term of cone, defined by ν i = coθ i and ψ = [0, π]. The phae function i rewritten a a reditribution function on matrix form, h(ν i,ν j ), which decribe the probability of cattering from cone ν i to cone ν j. The adding-doubling method firt aume that the reflectance R(ν i,ν j ) and tranmittance T(ν i,ν j ) from a thin, homogeneou, layer of infinite extenion are known. By juxtapoing two identical layer and umming the contribution from each, the reflectance and tranmittance from a layer twice a thick can be obtained. In thi fahion, the reflection and tranmiion propertie of a lab of arbitrary thickne can be calculated. In a imilar way, layer of different optical propertie can be added together, hence the name adding-doubling. The adding-doubling cheme conit of integrating dicrete reflection and tranmiion function. The numerical integration, quadrature, i therefore an important part. Different quadrature cheme are dicued in Ref. 40. Typically between 4 and 3 cone, equal to the number of quadrature point, are ued in adding-doubling calculation. The reflectance and tranmittance from the firt layer can be calculated in everal way. The mot widely ued method i diamond initialization, which aume that the radiance can be approximated by the average of the radiance at the top and bottom of the layer. The requirement for thi approximation to be valid i that the layer i optically thin. Furthermore, the RTE i written a time-independent, onedimenional, and with the angular component dicretized according to the cone approach: ν i L( x, νi ) + µ t L( x, νi ) = µ x N j= 1 w j [ h( ν, ν ) L( x, ν ) + h( ν, ν ) L( x, ν )] Solution of R and T for diamond initialization can be found in Ref. 40 and 41. i j j i j j (3.41) 31

32 The advantage of the adding-doubling method i that the olution are accurate for any combination of µ a, µ and g. Index mimatch between layer i alo handled correctly. The limitation of the method are that it i retricted to layered geometrie and uniform irradiation, that it doe not readily give light fluence inide the media, that each layer mut be homogeneou, and that the method i not time-reolved. Computer code for adding-doubling calculation, by Prahl, i available for download 4. Continuing with the dicretization approach, the next tep would be to olve the RTE for a full 3D geometry with N angular component. A even-flux method ha been ued in tiue optic 43. In thi method, the ix direction along the axe of a Carteian coordinate ytem are ued, and a eventh flux along the direction of the incident light beam i introduced. Uing only even angular component i not optimal in term of obtaining accurate reult, and higher number of N are needed for truly veratile dicrete ordinate model. Extenive development in dicrete ordinate ha been performed to model neutron tranport, but urpriingly little of thee reult have pilled over to light propagation. One reaon for thi may be that dicrete ordinate computation, up until recently, have required the ue of upercomputer to perform within reaonable time limit. Light propagation problem are actually impler than neutron propagation, becaue all photon move at a contant peed, which i not the cae for neutron. The principle of the dicrete ordinate method will be ketched briefly. To olve the RTE in a full 3D geometry, the patial coordinate need to be dicretized in addition to the angular direction. The patial dicretization can be done, e.g., uing the Crank-Nicolon method 44. A large number of trategie for dicretization have been invetigated (ee the review in Ref. 45). With thee dicretization, the RTE i tranformed into a et of coupled integro-differential equation. The next tep i to expand the phae function in a erie of Legendre polynomial P l (coθ), L l + 1 p(co θ) = b l P l (coθ). (3.4) 4π l= 0 The reader hould note that thi tep i identical to the procedure ued when deriving the diffuion approximation, a will be decribed in Sect In general term, the RTE ha now been converted to an equation ytem that can be written on the form 46 ( A B) L = Q, (3.43) where A and B are dicretized verion of the linear operator 3