kubelk munk theory
TRANSCRIPT
The Kubelka-Munk theory,applications and modifications
Frederic P.-A. Cortat
December 19, 2003
1
Overview
• Derivation of K-M equations
• Nature of K and S coefficients
• Applications and problems
• Revised K-M theory
• Applications and results
2
The Kubelka-Munk theory
3
Reflectance of layer on substrate
R0 and T are reflectance and transmittance oflayer. Set reflectance of substrate to Rg.Upward flux:
Jg = (I · T + Jg ·R0) ·Rg
Reflectance of layer:
R =I ·R0 + Jg · T
I= R0 +
T 2 ·Rg
1−Rg ·R0
R0 is reflectance of sample over ideally blackbackground (Rg = 0).
4
Reflectance of thin layer in medium
Reflectance and transmittance of layer are r0
and t. Therefore absorption is
a = 1− r0 − t
Change in i and j going from the n-th to then + 1-th layer
in+1 − in =(
1t− 1
)· in − r0
t· jn
jn+1 − jn =(
t− 1− r20
t
)· jn +
r0
t· in
Assumption 1: r0 and t are the same for i andj flux. Correct ⇔ angular distribution ofintensities are both equal.
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Reflectance of continuous medium
Assumption 2: sample may be treated ascontinuous medium.
Define ”scattering” coefficient S and”absorption” coefficient K:
S = limd→0
r0
d=
dR0
dx
K = limd→0
a
d= −dT
dx− dR0
dx
Taking limit d → 0 leads to K-M differentialequations
di
dx= −(K + S) · i + S · j
dj
dx= (K + S) · j − S · i
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Reflectance and transmittance values
Reflectance of infinite thick layer:
R∞ =1 + r2
0 − t2 −√
(1 + r20 − t2)2 − 4r2
0
2r0
≡ 1 +K
S−
√(1 +
K
S
)2
− 1
Solving K-M equations gives I and J , andtherefore
R0 =sinh(Z)
α · sinh(Z) + β · cosh(Z)
T =β
α · sinh(Z) + β · cosh(Z)
Z =√
K(K + 2S) ·X
where α := 1 +K
S, β :=
√a2 − 1.
R∞ = α− β =1
α + β
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The K-M coefficients K and S
K and S are defined in terms of transmittanceand reflectance of thin layer. Separate modelrequired to relate K and S to fundamentaloptical properties of material: absorption (ε)and scattering (σ) coefficients per unit pathlength.
Fractions absorbed and scattered overinfinitesimal distance du are ε · du and σ · du.For incident ray at angle θ, du = dx/cos(θ).For diffuse light, average path length is integralover angular distribution θ ∈ [0, 2π]:
⇒ K = 2 · ε
Assume light isotropically scattered. Only halfis scattered in upper half and contribute toreflectance:
⇒ S = σ
8
Applications of the K-M theory
9
Theory at test: predicted values of R∞
Checking accuracy of K-M theory is difficultbecause of restrictions imposed duringderivations.
Test conducted on values of R∞: exactagreement only for R∞ = 1 or R∞ = 0. Elseerror as large as 8%.
Albedo: a := σ/(σ + ε)
Large discrepancy disappointing andunexplained.
10
Improving the theory: modify K and S
Idea: separate K and S for forward (Ki, Si)and reverse (Kj , Sj) flux. Multi-flux analysisshows that angular distribution is indeed notthe same, even for ideal diffuse illuminationand isotropic scattering.
Result: coefficients can still be combined into asingle pair: K = 2 · ε , S = 0.75 · σ.Experiments showed that this is correct onlyfor weakly absorbing samples. For moreabsorbing samples, both ratios K/ε and S/σ
depend on ε and σ. This is in directdisagreement with K-M theory.
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Mathematical treatment of print-through
Print-through conventionally defined by
G = log(R∞/RG)
where R∞ is intrinsic reflectance of paper andRG is reflectance factor of reverse side of printwith opaque pad of paper as background.
Print-through can be divided into componentsrepresenting show through if no inkpenetration, contribution of ink penetration,and effect of oil separation from ink thatreduces opacity of paper.
G = GL + GP + GS
= log(R∞/RA) + log(RA/RQ) + log(RQ/RG)
RQ can be easily measured. What about RA?
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Mathematical treatment of print-through
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Mathematical treatment of print-through
Idea: RA = RX , the reflectance value of asingle sheet of unprinted paper placed overprinted surface.
RX is given by K-M theory:
R =R0 + Rg −R0 ·Rg
(R∞ + 1
R∞
)
1−R0 ·Rg
⇒ RX =R0 + RP −R0 ·RP
(R∞ + 1
R∞
)
1−R0 ·RP
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Math. treatment of ink penetration
Penetration depth Wp with respect to papergrammage.
Wp
W=
ln(B0/BZ)ln(B0)
, Bi :=1−Ri ·R∞1−Ri/R∞
K-M theory:
2b · S ·W = ln(
B
Bg
), 2b :=
1R∞
−R∞
RX for sheet with thickness W againstbackground RP ; RQ: thickness (W −WP )against background RP .
2b·S·W = ln(
BX
BP
), 2b·S(W−WP ) = ln
(BQ
BP
)
Wp
W=
ln(BX/BQ)ln(BX/BP )
=ln(B0 ·BP /BQ)
ln(B0)
Confirmed by computer simulations.
15
The revised Kubelka-Munk theory
16
Revised Kubelka-Munk theory
K-M theory successful and widely used inindustry. Nevertheless unable to explain somefindings ⇒ modifications necessary.
Motivations:
• K-M theory best for low absorption. Notgood at high absorption.
• Many restrictions/assumptions madeduring derivation
• K and S coefficients have no physicalmeaning
17
Light propagation in media
Mean path length free from absorption, respscattering: la(λ) := 1
ε(λ) , ls(λ) := 1σ(λ) .
Overall photon path:
la = 〈l〉 =N∑
n=1
〈| ~rn|〉 = N · ls
Mean square scattering distance:
〈~R2〉 =N∑
m=1
N∑n=1
〈 ~rm · ~rn〉 =N∑
n=1
〈 ~rn2〉 = N · l2s
R =√〈|~R|2〉 =
√la · ls
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Light propagation in media
Ratio between total path length and length ofcorresponding displacement:
µ :=laR
=√
lals
=√
σ
ε> 1
Including wave length dependence:
µ =
√σ(λ)ε(λ) σ(λ) > ε(λ)
1 otherwise
Because light absorption by the mediadepends on wavelength, µ can vary significantly(even for constant scattering).
In original K-M theory, scattering induced pathvariation was ignored: la = R ⇒ µ = 1.
19
Modified K-M equations
Average path length traversed by light goingdownward (upward defined similarly):
〈dl〉I = µ · dz
π/2∫
0
1I
∂I
∂φ
dφ
cos(φ)=: µ · αI · dz
Diffuse light: αI = 2. Collimated: αI = 1.Intensity variation after passing through dz:
(ε + σ) · I · 〈dl〉I = µ · (ε + σ) · I · αI · dz
New differential equations:
dI
dz= −µ · αI · (ε + σ) · I + µ · αJ · σ · J
dJ
dz= µ · αJ · (ε + σ) · J − µ · αI · σ · I
20
New K and S coefficients
For αI = αJ = α, new differential equationsreduce to original K-M equations iff
k = µ · α · ε , s =µ · α · σ
2
For diffuse light: k = 2µ · ε , s = µ · σ.
k and s depend on µ, itself depending on ε, σ
and λ:
k = µ · α · ε =
{α√
σ · ε σ(λ) > ε(λ)
α · ε otherwise
s =µ · α · σ
2=
α2
√σ3
ε σ(λ) > ε(λ)α·σ2 otherwise
k and s will change depending upon variationsin ε and σ ⇒ they are no properrepresentations of material properties.
21
Original K-M theory vs. revised theory
• K-M theory is particular case of revisedtheory
• In original K-M theory, k and s coefficientsare not physical quantities
• In revised theory, k and s are linkedelegantly to fundamental properties of thematerial
• Revised theory has broader range ofvalidity
22
Applications of the revised K-M theory
23
Application I: inks
Dye-based ink, subject to little scattering.Measurements → compute K-M scattering andabsorption powers → deduce µ → compute ε
and σ.
24
Application II: paper
Single sheet of paper, subject to strongscattering.
25
Application III: dyed paper
Assumptions: σp, εp and zp for dyed paperremain unchanged.
Additivity law:
εip · zp = ρ · εi · zi + εp · zp
σip · zp = ρ · σi · zi + σp · zp
K-M theory gives for k and s powers:
kip · zp = ρ · ki · zi + kp · zp
sip · zp = ρ · si · zi + sp · zp
Revised theory:
kip · zp = ρµip
µi
αip
αiki · zi +
µip
µp
αip
αpkp · zp
sip · zp = ρµip
µi
αip
αisi · zi +
µip
µp
αip
αpsp · zp
K-M theory is special case µip
µi= µip
µp= 1.
26
Application III: dyed paper
K-M theory: scattering dominated by paper.
Revised K-M theory: scattering dominated bypaper, but influence of µ factor:
sip · zp ≈ µip
µpsp · zp
µip(λ) ≈√
σp(λ) · zp
ρ · εi(λ) · zi + εp(λ) · zp
εi À εp ⇒ ρ > 0 lowers µip.Revised K-M theory accounts for drop ofscattering. Agree with experimentalobservations.
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Application III: dyed paper
K-M theory: absorption power increaseslinearly with ink concentration.
Revised K-M theory: absorption dominated byink, but influence of µ factor:
kip · zp ≈ ρµip
µiki · zi
µip depends on σp ⇒ µip
µiÀ 1 ⇒ absorbing
power of dyed paper larger than that of ink.Confirmed by measurements.
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