Aperture effects in a scattering medium a monte carlo phase space study of confocal imaging

.. .APERTURE EFFECTS IN A SCATTERING MEDIUM: A MONTE CARLO PHASE SPACE STUDY OF CONFOCAL IMAGING CAHYADI TJOKRO (B.Sc (Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING... the imaging information carried by the propagating light This is why the study of scattering processes in the scattering medium is of paramount importance in improving the current bio -imaging. .. observations in a scattering medium (µ s ) = 1.7 mm −1 with a mm annular aperture at ObPin and ObPout 58 Figure 5.27 Phase space profile of the photons on different planes of observations in a scattering

APERTURE EFFECTS IN A SCATTERING MEDIUM:

A MONTE CARLO PHASE SPACE STUDY OF CONFOCAL

IMAGING

CAHYADI TJOKRO

NATIONAL UNIVERSITY OF SINGAPORE

2006

APERTURE EFFECTS IN A SCATTERING MEDIUM:

A MONTE CARLO PHASE SPACE STUDY OF CONFOCAL

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Acknowledgement

The author would like to express a heartfelt gratitude to his Supervisor and Counselor, Prof Colin Sheppard for his guidance and insights throughout the period of this research project His undoubted trust and encouragement have always spurred the author to strive for the best of his ability in completing this project

Many thanks go to A/P Toh Siew Lok for his gracious support to the author, especially during the project transition period

The author would also like to thank his seniors, Mr Elijah Yew and Mr James Kah, for the inspiring and fruitful discussions around the research project

The author’s deepest appreciation goes to his best friend, Ms Natalia Filo Sutanto, who is always ready to encourage and brighten up the author’s life

Last but not least, the author would also like to thank God for His faithfulness and His blessings that the author is able to complete this project well within the time limit

5.1 Model 1: Cases without Aperture

5.2 Model 2: Cases with Aperture

5.3 Model 3: Cases with Koester’s Semi-Circular Aperture

5.4 Model 4: Cases with Annular Aperture

5.5 Scattering Statistic Analysis

A1.1 Header File

A1.2 Main Function File

A1.3 Operating File

A1.4 Input/Output Processing File

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74

78

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Summary

Research on the field of bio-imaging has experienced rapid progress due to its application in diagnosing human tumors non-invasively The usage of non-ionizing light presents trade-off between a safer imaging modality and a much limited depth and lateral resolution This limited performance is mainly due to the highly-scattering nature of most biological tissues that distorts the imaging information carried by the propagating light This is why the study of scattering processes in the scattering medium is of paramount importance in improving the current bio-imaging technology

Several theories such as elastic, quasi elastic and non-elastic scattering theory have been developed to describe different types of scattering processes Nevertheless these theories are only good in describing the scattering processes in a medium in which the probability of scattering is relatively low However, this is not the case for most biological tissues

A better theory to describe the light propagation and scattering processes is the one that is based on the linear transport equation The theory takes into account average optical properties of the medium such as the scattering coefficient, absorption coefficient, and anisotropy factor Nonetheless it is not always straightforward to solve the equation analytically, especially when the medium has a complex geometry and irregular boundary condition

One other way to solve the linear transport equation is by using Monte Carlo method This method is more geometrically flexible and statistically accurate via problem-

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solving through simulation governed by the stochastic principle based on the random sampling processes and the optical properties of the medium

In this study, such simulation method is employed to simulate light propagation inside

a scattering medium A new mechanism based on directional cosine is developed to simulate photon propagation through different planes within the context of a reflectance mode confocal microscope based on a 4-f optical system Several types of aperture such as circular, semi-circular and annular aperture are introduced into the microscope system The spatial distribution, spatial frequency distribution and phase space profile of the scattered photons under different circumstances are observed and discussed Statistics of the number of scattering events in different circumstances under different types of aperture are also discussed

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List of Figures

Figure 4.1 Diagram of the reflectance mode confocal microscope based on a 4-f optical system

considered in this study Light source with diameter D incident to an objective plane (ObP) before being focused to the focal plane (FoP) inside the turbid medium The reflected light is then transmitted through back-focal plane (BFP) and then to the detector at the image plane (ImP) 17 Figure 4.2 Schematic diagram of the 4-f optical system A photon enters the medium from its initial

position rin with ObPin with direction µObPin Due to the scattering nature of the medium, the photon

does not return to ObP at rin* Instead it reaches ObP at rout with direction vector µObBPout To get to

the BFP, the correct expression for µBFP is needed By a vectorial approach, we can get the expression

on the right top corner of the figure .20 Figure 4.3 Schematic diagram of the axial response of the confocal microscope due to a displacement of

an embedded reflecting mirror inside the medium The projected out-of-focus image due to the displacement is shown with yellow box The axial response of the microscope is taken as the ratio of pinhole area and the projected out-of-focus area in the image plane (ImP) .22 Figure 5.1 Spatial distribution profiles of the photons on different planes of observations in a non- scattering medium ( 1)

mm 0

mm 0

mm 7

=

s

µ 33 Figure 5.6 Phase space profile of the photons on different planes of observations in a scattering medium

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Figure 5.7 Spatial distribution profile of the photons on different planes of observations in a scattering medium ( 1)

mm 1

=

s

µ 35 Figure 5.8 Spatial frequency distribution profile of the photons on different planes of observations in a scattering medium ( µs = 5 1 mm−1) 36 Figure 5.9 Phase space profile of the photons on different planes of observations in a scattering medium

mm 7

mm 7

mm 7

mm 7

mm 7

mm 7

mm 7

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Figure 5.20 Spatial frequency distribution profile of the photons on different planes of observations in a scattering medium ( 1)

mm 7

Figure 5.23 Spatial frequency distribution profile of the photons on different planes of observations in a scattering medium ( 1)

mm 7

mm 7

Figure 5.26 Spatial frequency distribution profile of the photons on different planes of observations in a scattering medium ( 1)

mm 7

mm 7

Figure 5.29 Spatial frequency distribution profile of the photons on different planes of observations in a scattering medium ( µs = 1 7 mm−1) with a 3 mm annular aperture at ObPin and BFP 61 Figure 5.30 Phase space profile of the photons on different planes of observations in a scattering medium ( 1)

mm 7

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List of Tables

Table 4.1 Parameters of the Optical System 25 Table 4.2 Parameters of the Turbid Medium 25

of the object under observation Thus optical imaging is relatively much safer than other modalities using ionizing light This however has a consequence in the much limited depth and lateral resolution when imaging a hidden object in a turbid medium Due to the highly scattering property of turbid media, light experiences deviation on its path of travel inside a medium This leads to the distortion of the light spatial profile that carries the imaging information Techniques have been developed to tackle this problem One of the most widely known is the technique used in confocal microscope.3

In a confocal microscope, a small pinhole (point) aperture is used in the detection/image plane Similar ideas to use apertures/masks at different planes have also been studied elsewhere.3,4 These apertures act as a spatial filter/angular gate to block highly scattered out-of-focus light Interestingly, due to the physical nature of the optical system, the aperture filters light in different manners according to which plane in which it is placed

2

1.2 Objectives and Organization

In this study, investigations have been carried out to better understand the spatial dynamics of the scattered light from a medium at different planes of observations with and without the presence of an aperture By analyzing the spatial profile and the phase space of light at different planes, better appreciation of the scattered light is acquired, which will be valuable in improving the imaging technologies

This report is organized as follows: The second chapter provides background knowledge on light scattering theories and the different approach needed to describe scattering processes in a turbid medium The third chapter describes the Monte Carlo method used in this study; its sampling processes and algorithm The fourth chapter presents the experimental model; the optical system to be considered and the experimental models to achieve the objective The fifth chapter discusses photon profiles presented from the experiments The last chapter concludes the study by giving a summary and suggesting some future extension to the study

In general, scattering is an optical process whereby the direction and/or wavelength of the incident light wave are altered This alteration is due to collisions of light/photons with particles or centers of inhomogeneity in the medium Thus scattering can only occur in a heterogeneous medium Depending on the light wavelength and on the nature of the scattering medium such as its size, density, and the state of the centers of inhomogeneity, researchers have classified the scattering processes into several types:6

1 Elastic Light Scattering

This kind of scattering events changes only the direction of the light propagation The photon energy and the frequency of the initial light remain unchanged The scattering of light by particles that are small compared to the light wavelength (< 0.1 λ) can be described by the theory of Lord Rayleigh On the other hand, it is Mie theory that explains the scattering of light by large particles more accurately. 7

2 Quasi-Elastic Scattering

2.2 Light Scattering in Turbid Medium

The three scattering theories described in the previous section are good to describe the scattering processes in a medium in which the probability of scattering is relatively low.6 However their solutions become very complicated when it comes to describing scattering processes in a dense or turbid medium where not only multiple scattering events occur but also absorption processes where light is attenuated and transformed to another form of energy as it travels inside the medium Some examples of such medium are biological tissues, clouds, fogs, etc In such cases a different and more realistic approach is needed One such approach is the description of the light scattering processes inside a turbid medium as a propagation of photons using averaged properties of the medium according to the linear transport equation.11 The three most important optical properties in this approach are the scattering coefficient,

s

µ , the absorption coefficient, µa, and the anisotropy factor, g

The scattering coefficient and absorption coefficient determine the path-length of photons traveling in the medium between scattering and absorption events as follows:

where φ is the scattering azimuthal angle, p(θ,φ) is the probability distribution for the

cosine of the deflection angle cos(θ), and d ϖ = sin ( ) θ d θ d ϕ denotes the element of the solid angle A value of g = 1 implies that only forward scattering events take place A value of g = − 1 implies that only backward scattering events take place And g = 0 implies that the scattering is isotropic, which means the scattering angles are distributed evenly in all directions

According to the light transport equation, the light propagation inside tissue can be described by solving the Boltzmann equation:11

r L t

, , ,

, ,

,

1

4

Ω + Ω Ω Ω Ω

=

Ω +

Ω Ω

r r r

r

’) is the scattering phase function describing the scattering probability of photons traveling in direction

Ω r into a new direction Ω r ’, v is the speed of photons in the medium, and µt is the interaction coefficient of the medium which equals to (µs+ µa)

The left hand side of equation 2.4 refers to the flux of photons that escape from a certain volume element This loss consists of the change of the photon distribution function, loss of photons through the boundary surface, absorption, and scattering The right hand side of the equations refers to photons that enter the volume element These consists of gains from scattered photons from other locations and from a possible light source

The linear transport equation can be solved analytically or numerically However as the medium becomes more complex, the solution to the equation becomes complicated

as well In such a case, an approximation might need to be considered A linear approximation of the linear transport equation is called photon diffusion equation:11

t

t r t

r v t

r

r r

∇

⋅

∇

7

where Φ ( ) r r , t is the fluence rate defined in the position described by vector r r and in

time t, S ( ) r r , t is the fluence rate of the source, and D is the diffusion coefficient

( D = v 3 ( 1 − g ) µs)

The photon diffusion equation is an approximation based on the assumption that scattering dominates the processes, that the scattering events are totally isotropic in nature, that the volume is far from the boundary and any light source, and that photons reaching the photo-detector are random The photon diffusion equation is popular because it allows the calculation over various geometries Finite elements can also be employed to solve the equation

Another important and valuable approach in the linear transport paradigm is by using a Monte Carlo simulation Instead of trying to solve the transport equation either directly

or through approximation, the Monte Carlo method simulates the processes described

in the equation A more detailed discussion of this method is presented in the next chapter

The main advantage of using the Monte Carlo method are its ability to tackle complex geometries while its main disadvantage results from its statistical requirements, that leads to long computing time to obtain accurate result Nevertheless this disadvantage can be tackled by improved computing power or using more efficient algorithms

3.2 Random Sampling Processes

There are at least two random sampling processes required to simulate light migration

in a turbid medium The first one is the sampling free path length, l (which denotes the

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distance traveled between two adjacent scattering events) The second one is the sampling scattering angles; the deflection angle, θ , and the azimuthal angle, φ These sampling processes are not straightforward, firstly because they have their unique probability density functions that will be sampled by using uniform computer- generated random numbers Secondly, it is because they have different ranges of values that will be sampled by using random numbers in the range between 0 and 1

For a given random number χ1 of the variable χ in the interval (a,b) and its

normalized non-uniform probability density function p ( ) χ , χ can be sampled to a 1random number ξ that has a unit uniform probability density function with the 1following equation:

3.2.1 Sampling Free Path Length

The photon free path length, l, in the turbid medium is dependent upon the scattering

and absorption coefficients of the medium Its probability density function is given as follows:14

( ) l t ( l )

3.2.2 Sampling Scattering Angles

The scattering angles describe the direction by which the photon will propagate after a scattering or absorption event occurs There are the deflection angle, θ and the azimuthal angle, φ to be sampled statistically

For the sampled deflection angle, a probability density function of its cosine is used Jacques, et al.,18 suggested that the density function originally introduced by Henyey- Greenstein1,18-19 describes single scattering in tissue well The function is expressed as follows:

11

2 3 cos 2 2 1

2

2 1 cos

g

where g is the anisotropy factor mentioned in the previous chapter

Again, by applying equation (3.1) to equation (3.5), the sampled cosine deflection angle is:

1 2

, 0 if

2 2 1

2 1 2

g

g g

where ξ is a random generated number over the interval (0,1)

The case for azimuthal angle is rather straightforward since it is uniformly distributed over the interval 0 to2π The sampled azimuthal angle is then expressed as:

πξ

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3.3 The Algorithm

3.3.1 Initialization

Each photon is first assigned a 3-D coordinate vector r r and a 3-D directional vector

µ r to describe its spatial position and the direction of its propagation, respectively The

spatial position uses the Cartesian coordinate system (x,y,z) and the directional variable

uses the 3-D direction cosine system (µx, µy,µz) µx, µy, and µz are the cosines of the

angle between propagation vector r r with each respective x, y, and z axes In order to

represent a correct unit directional vector, it is required that the direction cosines are normalized as follows:

12 2 2

= + +

= µx µy µz

Each photon is also assigned with a unit weight in the beginning that represents the energy or intensity of the photon This weight variable will be used in the absorption, reflection, and in the detection processes It is also used to increase the accuracy and speed of the simulation

3.3.2 Photon Launching

The photon is first initialized at a position on the interface of the medium and its initial position vector, r r0, is set to be (0,0,0), making it as the reference point Assuming a

normally incident collimated beam, the photon direction vector, µ r0, is set to be (0,0,1)

which means that the photon is launched along the z-axis

−

=

t i

t i t

i

t i i

sp

R

α α

α α α

α

α α

2 2

2

tan

tan sin

14

) /

( current z

current current

Once the photon is at the new position, its weight is updated due to the absorption process as follows:

old t

a old

θ µ µ ϕ θ

µ

θ µ µ ϕ

µ ϕ µ µ

θ

µ

θ µ µ ϕ

µ ϕ µ µ

θ

µ

cos 1

cos

sin

cos 1

sin cos

sin

cos 1

sin cos

sin

2

2 2

z z new

y

y z x

z y new

y

x z y

z x new

15

3.3.3 The Exiting Photon

Every time a photon is about to travel to a new location, it is checked whether the new position is actually outside the medium In that case, the photon is transported to the boundary or interface of the medium with the ambience and the remaining step size is stored It is then verified whether there is an occurrence of total internal reflection (TIR) where the photon is reflected back due to its incident angle being larger than the

critical angle The Fresnel formula of equation (3.10) is applied to get the value of R(αi)

to decide the occurrence of such reflection with the following method:

Transmitte

R if

3.3.4 Photon Termination

A weight threshold is imposed for the photons For every photon with weight below the threshold value, the photon then undergoes a termination process In this report, a termination process based on Russian Roullete14 is adopted Instead of directly

This technique helps the simulation to follow the energy conservation principle and thus be more accurate by not discarding all photon with small, yet non-zero, weight

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4.1 System and Implementation

A reflectance-mode confocal microscope based on a 4-f optical system is considered

for the study (see Figure 4.1) The Monte Carlo method14 discussed in the previous chapter is implemented, and methods of ray tracing based on geometrical optics and a direction cosine system21 are proposed They are used for simulating the light propagation inside and outside the medium, respectively The ray tracing techniques employed, which include a incident beam focusing into the tissue and photon propagation through the planes of observation such as objective plane (ObP), focal plane, (FoP), back focal plane (BFP), and detector or image plane (ImP) are described

in more details below

Figure 4.1 Diagram of the reflectance mode confocal microscope based on a 4-f optical system considered in this study Light source with diameter D incident to an objective plane (ObP) before being focused to the focal plane (FoP) inside the turbid medium The reflected light is then transmitted through back-focal plane (BFP) and then to the detector at the image plane (ImP)

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Initially, the photon is launched from its assumed random position

) ,

0

0 0

The photon then propagates towards the medium through the objective plane (ObP) Once it arrives on the objective plane (ObPin), the direction of propagation of the photon is altered and focused to the focal plane inside the medium If the focal length

of the objective lens is fL1, the directional vector of the photon prior entering the medium is then set to be:

2 1 2 0 2 0 1 0

0

L OBPin

x

+ +

1

L ObP

ObP L

ObP ObP

out out

out out out

µ r is the direction vector of the back-reflected photon from the medium

prior entering the objective plane (ObP), f

ObP out

µ r is the direction vector to the focal

plane for photons at

out

ObP

r r , and µ r( z− ) is the direction vector parallel to the z-axis The

vectorial proof is illustrated in Figure 4.2

To analyze the spatial profile of photons on the image/detector plane, a slightly different expression than the one used by Schmitt15 is proposed Instead of using the

angle of emergence with respect to the x/y meridional plane, a more reliable and

general approach using directional vectors is preferred The expression is as follows:

out

ObP z L ObP

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Figure 4.2 Schematic diagram of the 4-f optical system A photon enters the medium from its initial

position rin with ObPin with direction µObPin Due to the scattering nature of the medium, the photon does

not return to ObP at rin* Instead it reaches ObP at rout with direction vector µObBPout To get to the BFP,

the correct expression for µBFP is needed By a vectorial approach, we can get the expression on the

right top corner of the figure

In this study, tracking of the photons is limited to photons which have experienced fewer than 100 scattering events To obtain a more enhanced back-reflection response

of the system, a mirror embedded inside the medium is simulated It has often been used to demonstrate the confocal effect and to measure the sectioning strength.3,15 This would mean every time the photon reaches or crosses the mirror line, it is then reflected back according to the same equation as equation (3.16)

4.2 Model Verification

There are two aspects involved in modeling light propagation in this study The first aspect is on the light propagation inside the medium where the light experiences multiple scattering events This aspect is simulated by using a Monte Carlo software

package developed by Wang, et al.14 The second aspect is on the light propagation outside the medium which includes the photon initialization, propagation into the

BFP (z=-d-fl)

ImP FoP

w/ mirror

Turbid

medium

ObPin/out(z=-d)

z f

ObP ObP

21

medium, as well as the photon propagation through various planes of observation outside the medium such as ObP, BFP, and ImP This aspect is simulated by using geometrical optics principle based on directional cosine system

Since the verification of the first aspect has been extensively reported elsewhere, 14 we will only discuss the verification of the second aspect in this report There are two ways to verify the second aspect: one is by verifying the axial response of the model against the geometrical optics theory and the other one is by verifying the spatial as well as the angular distribution profile of the reflected photons at various observation planes The first verification on the axial response will be done in this section, while the second verification will be discussed throughout Chapter 5 (especially Section 5.1)

To observe the axial response of the model, the reflecting mirror inside the medium is shifted and the detected photons, i.e., photons on the image plane which are within the pinhole area, are counted For comparison, a calculation to simulate this axial response

of a confocal microscope based on geometrical optics is derived (see Figure 4.3) Assuming the index-matched case, the portion of light which are detected on the detector plane within the pinhole area (i.e.: the axial response) can be calculated as follows:

2

2 2

2

2 (z)

image projected of

Area

pinhole of

Area )

d z

lens

lens

ph

lens lens

ph

π π

(4.5)

where z is the changes of the mirror position in z-axis, dph and dlens are the diameter

of the pinhole and objective lens respectively, and flens is the focal length of the objective lens

Figure 4.4 shows the corresponding axial response from both the modified Monte Carlo simulation (red circle) and the geometrical optics verification model (blue line)

ph

d was set to be 10 µm, dlens was 6 mm, and flens was 8.6 mm The microscope

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effectively then has a numerical aperture of around 0.33 As can be seen from Figure 4.4, the model agrees with the calculation based on geometrical optics

Figure 4.3 Schematic diagram of the axial response of the confocal microscope due to a displacement of

an embedded reflecting mirror inside the medium The projected out-of-focus image due to the displacement is shown with yellow box The axial response of the microscope is taken as the ratio of

pinhole area and the projected out-of-focus area in the image plane (ImP).

ImP

Incident light Projected Image Reflected light

z

Mirror

Figure 4.4 Comparison of Geometrical Optics theory with the Monte Carlo simulation results in the index-matched situation Inset is of the same data, zoomed at its peak to

see the agreement of the results

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4.3 Experimental Models

There are four different models considered in this study Model 1 focuses on the analysis of the spatial dynamics of the photons on different planes of observation Three cases with scattering coefficient of 0.0, 1.7, and 5.1 mm-1 are considered, labeled respectively as Cases 1, 2, and 3 Case 1 is intended to give the standard profile of the spatial photon distribution when there is no scattering event occurring Case 1 can be considered as the control case On the other hand, Case 2 and Case 3 are meant to show the dynamics and irregularity due to the scattering processes, especially their various distributions of the photons on different planes

Model 2 to Model 4 introduces apertures into the system of Model 1 Model 2 considers the effect of placing a circular aperture on the objective plane (ObP), the back focal plane (BFP), and the image plane (ImP) separately In this model only Case

2 are chosen for the study and only the profiles of photons on ObP, BFP, and ImP are studied

Model 3 studies the effect of placing semi-circular half apertures, which was introduced by Koester23 The first analysis for Model 3 is done when semi-circular apertures are placed on ObPin and ObPout After which, the analysis is carried out for the case in which the semi-circular apertures are placed on ObPin and BFP Only Case

2 is considered in this section

Annular apertures4,22 are introduced in Model 4 Its treatment follows that of Model 3

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In all the experimental models and cases, every photon is tracked in order to obtain information on its spatial position, its spatial frequency (which is represented by the direction cosines µx, µy, and µz), its scattering history, and lastly its remaining weight

This information is then used to plot: the spatial distribution profile in xy, the spatial

frequency profile in µxµy, the phase space profile which combines both spatial position and frequency, and the scattering events histogram These plots will be discussed in the next chapter

The parameters of both the optical system (Table 4.1) as well as the turbid medium (Table 4.2) in this study are set to be similar to the parameters used by Schmitt.15 The selected anisotropy factor of 0.92 refers to the dominant forward scattering events which is common in biological tissue with micro-spheres particle of 1.05 µm in diameter The scattering coefficient of 1.7 and 5.1 mm-1 corresponds to the optical depth of 1 and 3, respectively

Lens focal length

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5.1 Model 1: Cases without Aperture

Figure 5.1 to Figure 5.3 show the photon profiles in the case where no scattering occurs The photons are indicated with black dots From Figure 5.1, it can be seen that the spatial distribution of the photons in ObPin, ObPout, and BFP are all similar, i.e.: a circle field with diameter of 6 mm (diameter of lens) This means the photons are distributed inside the spatial boundary limit set by the lens and this is consistent with what is expected This is easy to understand for the case of photons at ObPin and ObPout planes where the lens is placed, but it is also true for photons at BFP (see Figure 4.2) On the other hand, the spatial distribution profiles at FoP and ImP are just a single point, as also expected for light perfectly focusing into the focal plane, FoP, or into its conjugate, the image plane, ImP

In Figure 5.2, the spatial frequency profiles of the photons are also in a circular shape pattern, except for the spatial frequency profile of photons at BFP which is just a single point The circular shape shown in the profiles at ObPin, ObPout, FoP, and ImP indicates that the photons are either directed to or directed from all directions

depending on the sign of their respective µz On the other hand, the single dot in the spatial frequency profile of photons at BFP means that the photons are propagating

only in the z-direction with µx = 0 , µy = 0

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Figure 5.1 Spatial distribution profiles of the photons on different planes of observations in a scattering medium ( µs = 0 0 mm−1) The photons are illustrated by black dots

non-28

Figure 5.2 Spatial frequency distribution profile of the photons on different planes of observations in a non-scattering medium ( µs = 0 0 mm−1) The photons are illustrated by black dots

A more interesting observation can be seen in the phase space profiles of the photons

(Figure 5.3) A phase space profile provides not only the spatial information (x, y

component) of a given set of photons, but it also provides the directional information

(spatial frequency µx, µy) A note is to be taken here that without loss of generality and

29

due to visualization constraint, only the 2D sections (the x, µx profile) of the full 4D

plots of the x, y, µx, µy components of the photons is used in this report

The phase space profile of photons at ObPin (top left, Figure 5.3) explains that the

photons are distributed uniformly in a circle centered at the z-axis and directed toward

a single point This is obtainable from the gradient of the line in the profile as well as

the magnitude of the axial components, x and µx The negative gradient of the line in this case means that the photons are converging to a point The magnitude of ± 0.313

in the spatial frequency, µx, means that the point of convergence is located at the focus point ( z = fl − d = 0 564 mm ) and the magnitude of ± 3 in the spatial position, x,

means that the photons are propagating from a circle of radius 3 mm Since the plots of

y, µy resemble those of x,µx, it is sufficient to analyze either one in the context of this

report

A brief note should also be presented here to explain the non-uniformity found on the width of the phase space profiles in Figure 5.3 It is basically due to how the positional and directional variables were set up From equation (4.1) and (4.2), we then have:

2

2 1

2

2 cos 2

L

ObP

x

f D

From this equation it is clear that the non-uniformity is contributed by the cosine terms and the two random numbers used Basically there is a nonlinearity introduced by the large value for the angles, and the thickness of the line in Figure 5.3 is caused by the

variations in the y,µy coordinates These effects are small We will see later that they