Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

The photon transport equation for turbid biological media with spatially varying isotropic refractive index

Malin Premaratne, Erosha Premaratne, and Arthur James Lowery

Open Access Open Access

Abstract

Using the principle of energy conservation and laws of geometrical optics, we derive the photon transport equation for turbid biological media with spatially varying isotropic refractive index. We show that when the refractive index is constant, our result reduces to the standard radiative transfer equation and when the medium is lossless and free of scattering to the well known geometrical optics equations in refractive media.

©2005 Optical Society of America

Full Article  |  PDF Article
More Like This
Scaling property of the diffusion equation for light in a turbid medium with varying refractive index

Margarita L. Shendeleva and John A. Molloy
J. Opt. Soc. Am. A 24(9) 2902-2910 (2007)

Radiative transfer in a turbid medium with a varying refractive index: comment

Margarita L. Shendeleva
J. Opt. Soc. Am. A 21(12) 2464-2467 (2004)

Validity conditions for the radiative transfer equation

Luis Martı́-López, Jorge Bouza-Domı́nguez, Jeremy C. Hebden, Simon R. Arridge, and René A. Martı́nez-Celorio
J. Opt. Soc. Am. A 20(11) 2046-2056 (2003)

View More...

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)
Fig. 1.
Fig. 1. Transport of an infinitesimally small photon packet with phase space volume, V S 0 , along an infinitesimally small ray tube surrounding a central light ray.

Download Full Size | PDF

Fig. 2.
Fig. 2. Details of the stationary and moving coordinate systems

Download Full Size | PDF

Equations (54)

Equations on this page are rendered with MathJax. Learn more.

Ω ̂ = d r d s = 1 − μ 2 cos ⁡ ( φ ) x ̂ + 1 − μ 2 sin ⁡ ( φ ) y ̂ + μ z ̂
∫ V 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
∫ δ V s + Δ s 𝓛 ( x , y , z , μ , φ , t + Δ t ) d x d y d z d μ d φ − ∫ δ V s 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
= Δ s ∫ δ V s ( μ a + μ s ) 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
+ Δ s ∫ δ V s μ s ( ∫ − 1 1 ∫ 0 2 π f ( μ , φ , μ ̀ , φ ̀ ) 𝓛 ( x , y , z , μ ̀ , φ ̀ , t ) d μ ̀ d φ ̀ ) d x d y d z d μ d φ
+ Δ s ∫ δ V s ε ( x , y , z , μ , φ , t ) d x d y d z μ d φ + O ( Δ s 2 )
∫ δ V s + Δ s 𝓛 ( x , y , z , μ , φ , t + Δ t ) d x d y d z d μ d φ − ∫ δ V s 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
= ∫ δ V s 0 [ ( 𝓛 J ) ∣ s + Δ s , t + Δ t − ( 𝓛 J ) ∣ s , t ] d x 0 d y 0 d z 0 d μ 0 d φ 0
( 𝓛 J ) ∣ s + Δ s , t + Δ t − ( 𝓛 J ) ∣ s , t = J n c ∂ 𝓛 ∂ t Δ s + J ∂ 𝓛 ∂ s Δ s + L ∂ J ∂ s Δ s + O ( Δ s 2 )
J = ∣ ∂ x ∂ x 0 ∂ x ∂ y 0 ∂ x ∂ z 0 ∂ x ∂ μ 0 ∂ x ∂ φ 0 ∂ y ∂ x 0 ∂ y ∂ y 0 ∂ y ∂ z 0 ∂ y ∂ μ 0 ∂ y ∂ φ 0 ∂ z ∂ x 0 ∂ z ∂ y 0 ∂ z ∂ z 0 ∂ z ∂ μ 0 ∂ z ∂ φ 0 ∂ μ ∂ x 0 ∂ μ ∂ y 0 ∂ μ ∂ z 0 ∂ μ ∂ μ 0 ∂ μ ∂ φ 0 ∂ φ ∂ x 0 ∂ φ ∂ y 0 ∂ φ ∂ z 0 ∂ φ ∂ μ 0 ∂ φ ∂ φ 0 ∣
J = ∑ σ sgn ( σ ) J 1 , σ ( 1 ) J 2 , σ ( 2 ) J 3 , σ ( 3 ) J 4 , σ ( 4 ) J 5 , σ ( 5 )
J = ∑ σ sgn ( σ ) ∏ j = 1 5 ∂ ϑ ( j ) ∂ ϑ 0 ( σ ( j ) )
∂ J ∂ s = ∑ v ∈ { x 0 , y 0 , z 0 } ∑ σ sgn ( σ ) ( ∑ k = 1 5 ∂ Ω v ∂ ϑ ( k ) ∂ ϑ ( k ) ∂ v ) ∏ j = 1 ϑ 0 ( j ) ≠ v 5 ∂ ϑ ( j ) ∂ ϑ 0 ( σ ( j ) )
+ ∑ v ∈ { μ 0 , φ 0 } ∑ σ sgn ( σ ) ( ∑ k = 1 5 ∂ ϑ ( k ) ∂ v ∂ ∂ ϑ ( k ) ( ∂ v ∂ s ) ) ∏ j = 1 ϑ 0 ( j ) ≠ v 5 ∂ ϑ ( j ) ∂ ϑ 0 ( σ ( j ) )
∂ J ∂ s = ( ∂ Ω x ∂ x + ∂ Ω y ∂ y + ∂ Ω z ∂ z + ∂ ∂ μ ( ∂ μ ∂ s ) + ∂ ∂ φ ( ∂ φ ∂ s ) ) J
∇ r · V = ∂ V x ∂ x + ∂ V y ∂ y + ∂ V z ∂ z
∂ J ∂ s = J ∇ r · Ω ̂ + ( ∂ ∂ μ ( ∂ μ ∂ s ) + ∂ ∂ φ ( ∂ φ ∂ s ) ) J
∇ r · Ω ̂ = 1 R 1 ( s ) + 1 R 2 ( s )
∂ ( n Ω ̂ ) ∂ s = ∇ r n
∂ Ω ̂ ∂ s = ∇ r n n − 1 n d n d s Ω ̂
μ ̂ = μ cos ⁡ ( φ ) x ̂ − μ sin ⁡ ( φ ) y ̂ + 1 − μ 2 z ̂
φ ̂ = − sin ⁡ ( φ ) x ̂ + cos ⁡ ( φ ) y ̂
∂ Ω ̂ ∂ s = 1 1 − μ 2 ∂ μ ∂ s μ ̂ + 1 − μ 2 ∂ φ ∂ s φ ̂
∂ μ ∂ s = 1 − μ 2 ∇ r n · μ ̂ n
∂ φ ∂ s = ∇ r n · φ ̂ n 1 − μ 2
∂ μ ̂ ∂ μ = − Ω ̂ 1 − μ 2
∂ φ ̂ ∂ φ = μ μ ̂ − 1 − μ 2 Ω ̂
∂ ∂ μ ( ∂ μ ∂ s ) = − μ 1 − μ 2 ∇ r n · μ ̂ n − ∇ r n · Ω ̂ n
∂ ∂ φ ( ∂ φ ∂ s ) = − μ 1 − μ 2 ∇ r n · μ ̂ n − ∇ r n · Ω ̂ n
∂ J ∂ s = ( 1 R 1 ( s ) + 1 R 2 ( s ) − 2 ∇ r n · Ω ̂ n ) J
∂ J ∂ s = ( 1 R 1 ( s ) + 1 R 2 ( s ) − 2 n d n d s ) J
∫ δ V s + Δ s 𝓛 ( x , y , z , μ , φ , t + Δ t ) d x d y d z d μ d φ − ∫ δ V s 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
= Δ s ∫ δ V s 0 [ n c ∂ 𝓛 ∂ t + ( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 + n 2 ∂ ∂ s ( 𝓛 n 2 ) ] J d x 0 d y 0 d z 0 d μ 0 d φ 0 + O ( Δ s 2 )
∫ δ V s + Δ s 𝓛 ( x , y , z , μ , φ , t + Δ t ) d x d y d z d μ d φ − ∫ δ V s 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
= Δ s ∫ δ V s [ n c ∂ 𝓛 ∂ t + ( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 + n 2 ∂ ∂ s ( 𝓛 n 2 ) ] d x d y d z d μ d φ + O ( Δ s 2 )
∫ δ V s [ n c ∂ 𝓛 ∂ t + ( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 + n 2 ∂ ∂ s ( 𝓛 n 2 ) ] d x d y d z d μ d φ
− ∫ δ V s ( ( μ a + μ s ) 𝓛 + ε ) d x d y d z d μ d φ
− ∫ δ V s μ s ( ∫ − 1 1 ∫ 0 2 π f ( μ , φ , μ ̀ , φ ̀ ) 𝓛 ( x , y , z , μ ̀ , φ ̀ , t ) d μ ̀ d φ ̀ ) d x d y d z d μ d φ = 0
n ( r ) c ∂ 𝓛 ( r , Ω ̂ , t ) ∂ t + ( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 ( r , Ω ̂ , t ) + n 2 ( r ) ∂ ∂ s ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) )
= ( μ a ( r ) + μ s ( r ) ) 𝓛 ( r , Ω ̂ , t ) + ε ( r , Ω ̂ , t ) + μ s ( r ) ∫ 4 π f ( Ω ̂ , Ω ̀ ) 𝓛 ( r , Ω ̀ , t ) d Ω ̀
∇ r 𝒮 = n ( r ) Ω ̂
n 2 ( r ) ∂ ∂ s ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) )
= n 2 ( r ) ∂ r ∂ s · ∇ r ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) ) + n 2 ( r ) ∂ Ω ̂ ∂ s · ∇ Ω ̂ ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) )
∇ Ω ̂ ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) ) = μ ̂ 1 − μ 2 ∂ ∂ μ ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) ) + φ ̂ 1 − μ 2 ∂ ∂ φ ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) )
n 2 ( r ) ∂ ∂ s ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) )
= Ω ̂ · ∇ r 𝓛 ( r , Ω ̂ , t ) + 1 n ( r ) ∇ r n ( r ) · ∇ Ω ̂ 𝓛 ( r , Ω ̂ , t ) − 2 n ( r ) Ω ̂ · ∇ r n ( r )
n ( r ) c ∂ 𝓛 ( r , Ω ̂ , t ) ∂ t + Ω ̂ · ∇ r 𝓛 ( r , Ω ̂ , t ) + 1 n ( r ) ∇ r n ( r ) · ∇ Ω ̂ 𝓛 ( r , Ω ̂ , t )
+ ( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 ( r , Ω ̂ , t ) − 2 n ( r ) Ω ̂ · ∇ r n ( r )
= ( μ a ( r ) + μ s ( r ) ) 𝓛 ( r , Ω ̂ , t ) + ε ( r , Ω ̂ ) + μ s ( r ) ∫ 4 π f ( Ω ̂ , Ω ̀ ) 𝓛 ( r , Ω ̀ , t ) d Ω ̀
n 0 c ∂ 𝓛 ( r , Ω ̂ , t ) ∂ t + Ω ̂ · ∇ r 𝓛 ( r , Ω ̂ , t )
= ( μ a ( r ) + μ s ( r ) ) 𝓛 ( r , Ω ̂ , t ) + ε ( r , Ω ̂ , t ) + μ s ( r ) ∫ 4 π f ( Ω ̂ , Ω ̀ ) 𝓛 ( r , Ω ̀ , t ) d Ω ̀
( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 ( r , Ω ̂ ) + n 2 ( r ) d d s ( 𝓛 ( r , Ω ̂ ) n 2 ( r ) ) = 0
𝓛 ( r , Ω ̂ ) = n 2 ( r ) n 2 ( r 0 ) 𝓛 0 exp ( − ∫ s 0 s ( 1 R 1 ( s ) + 1 R 2 ( s ) ) d s )
𝓛 ( r , Ω ̂ ) = n 2 ( r ) n 2 ( r 0 ) 𝓛 0
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use