1942
J. Opt. Soc. Am. A / Vol. 21, No. 10 / October 2004
Yang et al.
Revised Kubelka–Munk theory. II. Unified
framework for homogeneous
and inhomogeneous optical media
Li Yang, Björn Kruse, and Stanley J. Miklavcic
Campus Norrköping (ITN), Linköping University, S-601 74, Norrköping, Sweden
Received January 15, 2004; revised manuscript accepted April 14, 2004; accepted May 5, 2004
We extend the applicability of the recently revised Kubelka–Munk (K–M) theory to inhomogeneous optical
media by treating inhomogeneous ink penetration of the substrate. We propose a method for describing light
propagation in either homogeneous or inhomogeneous layers using series representations for the K–M scattering and absorption coefficients as well as for intensities of the upward and downward light streams. The
conventional and matrix expressions for spectral reflectance and transmittance values of optically homogeneous media in the K–M theory are shown to be special cases of the present framework. Three types of ink
distribution—homogeneous, linear, and exponential—have been studied. Simulations of spectral reflectance
predict a depression of reflectance peaks and reduction of absorption bands characteristic of hue shifts and
significant reduction of saturation and, in turn, color gamut. © 2004 Optical Society of America
OCIS codes: 000.3860, 120.5700, 120.7000, 290.7050.
1. INTRODUCTION
The original theory of Kubelka–Munk (K–M) was developed for light propagation in parallel colorant layers of infinite extension in the xy plane.1,2 The fundamental assumptions of the K–M theory were that the layer was
homogeneous and that the light distribution inside the
layer was completely diffuse. From these assumptions,
light propagation in the layer was simply represented by
two diffuse light fluxes through the layer, one upward, the
other simultaneously downward. After its introduction
in the 1930s the K–M theory was extended by removing
some of the original assumptions. Among others, a correction for boundary reflection at the interface between
two adjacent media was introduced by Saunderson,3 i.e.,
the so-called Saunderson correction. Kubelka himself
also made an attempt to extend the applicability of the
theory to optically inhomogeneous samples.4 However,
this extension was applicable only to the special case of
inhomogeneous media in which the ratio of the absorption
to the scattering was constant. A more generalized approach based on regular perturbation theory for inhomogeneous optical media was proposed by Mandelis and
Grossman.5 With this model, they computed the reflectance and transmittance of media layers with an exponential inhomogeneity.
The present work reflects the continuing efforts toward
improving matters by extending the applicability of the
revised K–M theory of part I of this paper6 to inhomogeneous media systems, as occurs for instance with an inhomogeneous ink distribution in a paper substrate incident
to ink distribution in paper. This work necessarily
complements the work of Mandelis and Grossman as the
dependence of the K–M scattering and absorption coefficients S and K on the ink distribution ␳ (z) is more complicated than in the original theory. Consequently, special care has been taken in the derivation of formulas to
1084-7529/2004/101942-11$15.00
ease numerical implementation. Moreover, the present
work provides a unified framework for modeling either
homogeneous or inhomogeneous media distributions.
This paper consists of five main sections. Section 2
provides a brief description of the revised K–M theory together with a collection of boundary conditions. Expressions for the spectral reflectance and transmittance of a
media layer having arbitrary types of media distribution,
with or without backing, are derived in Section 3. Section 4 describes examples of three types of ink penetration: homogeneous, linear, and exponential. Section 5
consists of simulations and discussions. In Section 6 we
summarize our results.
2. DIFFERENTIAL EQUATIONS AND
BOUNDARY CONDITIONS
Figure 1 is a schematic diagram of ink penetration of paper in which the ink penetration begins from the surface
of the substrate at z ⫽ D and ends at z ⫽ 0. In the case
of inhomogeneous ink dispersion, the intrinsic scattering
and absorption coefficients of the ink–paper mixture are
functions of the depth of ink penetration z; i.e., s ⫽ s(z)
and a ⫽ a(z), depending on the distribution of ink concentration. Consequently, the K–M scattering and absorption coefficients S and K of the ink–paper mixture
in the differential equations
dI
dz
dJ
dz
⫽ 共 S ⫹ K 兲 I ⫺ SJ,
⫽ ⫺共 S ⫹ K 兲 J ⫹ SI
(1)
are also z-dependent, because
K 共 z 兲 ⫽ ␮␣ a,
© 2004 Optical Society of America
S 共 z 兲 ⫽ ␮␣ s/2.
(2)
Yang et al.
Vol. 21, No. 10 / October 2004 / J. Opt. Soc. Am. A
Fig. 1. Schematic diagram of a linear ink distribution (with
backing). Ink penetration begins from the surface of the substrate at z ⫽ D and ends at z ⫽ 0. The remaining part of the
paper sheet (below z ⫽ 0) acts as a backing for the inked part.
In Eqs. (2) ␮ is the scattering-induced-path-variation
(SIPV) factor, which depends on the relative strength between scattering and absorption, i.e.,
␮共 z 兲 ⫽
再
关 s 共 z 兲 /a 共 z 兲兴
1,
1/2
s共 z 兲 ⭓ a共 z 兲
otherwise
,
(3)
while ␣ is a factor describing light distribution in the mixture and is defined by Eq. (16) in part I.6 The main task
of this paper is then to solve the differential equations
with S ⫽ S(z) and K ⫽ K(z).
The layer of the ink–paper mixture shown in Fig. 1 interfaces with air and the remainder of the paper sheet at
z ⫽ D and z ⫽ 0; respectively. Assuming that the layer
is in optical contact with the remainder of the paper with
a backing of reflectance R g at z ⫽ 0, one obtains
J a共 0 兲 ⫽ I a共 0 兲 R g .
(4)
If the external and internal surface reflections at the z
⫽ D interface are r 0 and r 1 , respectively, as shown in the
figure, one may then deduce the following boundary conditions at the z ⫽ D interface7:
I b共 D 兲 ⫽ I 0共 1 ⫺ r 0 兲 ⫹ J b共 D 兲 r 1 ,
I a 共 D 兲 R ⫽ I a 共 D 兲 r 0 ⫹ J b 共 D 兲共 1 ⫺ r 1 兲 ,
Figure 1 shows schematically the extent of ink penetration into a paper sheet. Let Z ⫽ z/D and let D denote
the thickness of the ink-penetrated layer extending to the
point where the ink concentration is essentially zero.
Then the ink penetration begins from the upper surface of
the paper sheet Z ⫽ 1 or z ⫽ D and ends at Z ⫽ 0. The
remaining part of the sheet remains clear, forming a
background for the ink-penetrated part. Except in the
case of a complete print-through, the ink penetrates only
partially into the paper sheet. Here the thickness D will
be only a fraction of the total paper thickness.
Suppose that the ink distribution as described by the
ink density relative to that of a printed ink layer that
does not mix with paper fibers (caused by ink penetration)
varies only in the z direction, i.e.,
␳共 Z 兲 ⫽ ␳
z
D
共 0 ⬍ z ⭐ D 兲.
,
(6)
A. K–M Scattering and Absorption Coefficients
Suppose the K–M coefficients of absorption and scattering are K p , S p and K i , S i for paper and ink, respectively.
By measuring the spectral reflectance properties of a
piece of paper against a black and a white backing, one
obtains two sets of experimental spectral reflectance values. K p and S p can then be computed by employing the
K–M theory. Similarly, one obtains another two sets of
spectral reflectance values by measuring the once and
twice printed ink layers on overhead-projector film (with
a white backing9).
According to Eq. (2) the scattering and absorption coefficients of the ink s i , a i and the paper a p , s p can be computed by
a i ⫽ K i / ␮ i␣ i ,
s i ⫽ 2S i / ␮ i ␣ i ,
a p ⫽ K p / ␮ p␣ p ,
s p ⫽ 2S p / ␮ p ␣ p ,
where
␮ i共 ␭ 兲 ⫽
3. SERIES SOLUTION OF THE
DIFFERENTIAL EQUATIONS
For inkjet printing, the absorption of ink constituents by
a paper medium is driven by the thermodynamic interaction between the ink and the paper, manifested by capillary forces and chemical diffusion gradients. Capillary
pressure is acknowledged to be the main driving force in
the offset-ink-oil transport in a typical porous papercoating structure. However, with increasing latex content, the diffusion-driven transport of ink chemicals into
the latex counterpart of the coating layer becomes
important.8 The different mechanisms driving ink penetration result in different forms of ink distribution in the
paper. Coping with this degree of variation demands a
framework that can deal with any form of ink distribution
in a unified manner.
冉冊
The form taken by the ink distribution in the paper depends on the rheological properties of the ink, the surface
and bulk properties of the paper, and their interactions.
(5)
where R is the reflectance of the medium.
In Eqs. (4) and (5) the subscripts a and b denote the
corresponding values (a) above and (b) below the interface. Therefore I a (D) ⫽ I 0 , as shown in the figure.
1943
再冋
2S i 共 ␭ 兲
K i共 ␭ 兲
册
1/2
,
1,
and
␮ p共 ␭ 兲 ⫽
再冋
2S p 共 ␭ 兲
K p共 ␭ 兲
1,
2S i 共 ␭ 兲 ⭓ K i 共 ␭ 兲
otherwise
册
1/2
,
2S p 共 ␭ 兲 ⭓ K p 共 ␭ 兲
otherwise
are the SIPV factors of the ink and the paper, respectively.
In the case of ink penetration, the ink penetrates into
the porous structure of the paper, forming an ink–paper
mixture. The tiny size of the dye molecules and the relatively small amount of dye present suggest that the structure of the paper remains the same after printing.
1944
J. Opt. Soc. Am. A / Vol. 21, No. 10 / October 2004
Yang et al.
Applying the additivity law, the absorption and scattering
coefficients of the ink–paper composite a ip , s ip can then
be expressed as
␮ ip 共 ␭ 兲 ⫽
a ip ⫽ a p ⫹ ␳ 共 Z 兲 a i ,
冦
冋
2h 共 S p 共 ␭ 兲 , S i 共 ␭ 兲 , Z 兲
h 共 K p共 ␭ 兲 , K i共 ␭ 兲 , Z 兲
册
1/2
,
2h 共 S p , S i , Z 兲 ⬎ h 共 K p , K i , Z 兲
1,
.
otherwise
s ip ⫽ s p ⫹ ␳ 共 Z 兲 s i ,
where ␳ (Z) is the ink concentration relative to that of the
pure ink layer. The SIPV factor of the ink–paper mixture is then computed by6
␮ ip 共 ␭ 兲 ⫽
再冋
s p共 ␭ 兲 ⫹ ␳ 共 Z 兲 s i共 ␭ 兲
a p共 ␭ 兲 ⫹ ␳ 共 Z 兲 a i共 ␭ 兲
册
Consequently, the K–M scattering and absorption coefficients for the ink–paper mixture can be expressed completely in terms of its components, i.e.,
1/2
s ip 共 ␭ 兲 ⬎ a ip 共 ␭ 兲
,
1,
.
otherwise
K ip 共 ␭ 兲 ⫽
Clearly, the SIPV in the ink–paper mixture depends on
the ink concentration ␳ (Z). According to Eq. (2), the
K–M scattering and absorption coefficients are then
K ip 共 Z 兲 ⫽ ␮ ip ␣ ip a ip
⫽
␮ ip ␣ ip
␮p ␣p
Kp ⫹ ␳共 Z 兲
␮ ip ␣ ip
␮i ␣i
S ip 共 ␭ 兲 ⫽
Ki
⫽ ␮ ip h 共 K p , K i , Z 兲 ,
(7)
冦
关 2h 共 S p , S i , Z 兲 h 共 K p , K i , Z 兲兴 1/2,
冦
2h 2 共 S p , S i , Z 兲 /K ip ,
2h 共 S p , S i , Z 兲
⬎ h共 Kp , Ki , Z 兲
h共 Kp , Ki , Z 兲,
(10)
.
(11)
otherwise
2h 共 S p , S i , Z 兲
⬎ h共 Kp , Ki , Z 兲
h共 Sp , Si , Z 兲,
,
otherwise
S ip 共 Z 兲 ⫽ ␮ ip ␣ ip s ip /2
⫽
␮ ip ␣ ip
␮p ␣p
Sp ⫹ ␳共 Z 兲
␮ ip ␣ ip
␮i ␣i
Si
⫽ ␮ ip h 共 S p , S i , Z 兲 ,
(8)
where
h 共 x, y, Z 兲 ⫽
1 ␣ ip
␮p ␣p
x⫹
␳ 共 Z 兲 ␣ ip
␮i
␣i
y,
(9)
with x ⫽ S p , K p and y ⫽ S i , K i . ␣ i , ␣ p , and ␣ ip are
quantities related to the light distribution in the ink, the
paper, and the ink–paper mixture, respectively. Similarly ␮ i , ␮ p , and ␮ ip are the SIPV factors of the ink, the
paper, and the ink–paper mixture, respectively. These
quantities are generally different for the different media
(ink, paper, and ink–paper) because of different scattering and absorption properties. Equations (7) and (8) are
expressions describing the relationships between the
K–M scattering and absorption coefficients for the ink–
paper mixture and those of the ink and paper. The additivity results advocated in earlier publications7,10 hold
only in an extreme, special case: When ␮ ip ⫽ ␮ p ⫽ ␮ i
and ␣ ip ⫽ ␣ p ⫽ ␣ i .
Since
s ip /a ip ⫽ 2S ip /K ip ⫽ 2h 共 S p , S i , Z 兲 /h 共 K p , K i , Z 兲 ,
the SIPV factor ␮ ip can be expressed as
The dependencies of the K–M scattering and absorption coefficients for the ink–paper mixture, K ip , S ip on
the form of the ink penetration ␳ are generally nonlinear
as indicated by Eqs. (10) and (11), unless s ip ⬍ a ip and
␮ ip ⫽ 1. This is in sharp contrast to previous theoretical
predictions based on the original K–M theory stating that
K ip and S ip are linear superpositions of the K–M coefficients of ink and paper K i , K p and S i , S p ,
respectively.10–12
For simplicity we denote the K–M absorption and scattering coefficients for the ink–paper mixture as functions
f and g of the Z coordinate:
f 共 Z 兲 ⫽ K ip 共 ␭ 兲
⫽
冦
关 2h 共 S p , S i , Z 兲 h 共 K p , K i , Z 兲兴 1/2,
2h 共 S p , S i , Z 兲
⬎ h共 Kp , Ki , Z 兲
h共 Kp , Ki , Z 兲,
,
(12)
.
(13)
otherwise
g 共 Z 兲 ⫽ S ip 共 ␭ 兲
⫽
冦
2h 2 共 S p , S i , Z 兲 /K ip ,
2h 共 S p , S i , Z 兲
⬎ h共 Kp , Ki , Z 兲
h共 Sp , Si , Z 兲,
otherwise
Yang et al.
Vol. 21, No. 10 / October 2004 / J. Opt. Soc. Am. A
Here it is convenient to expand the K–M absorption f(Z)
and scattering g(Z) coefficients in Taylor series in the vicinity of Z ⫽ 0, i.e.,
1
f共 Z 兲 ⫽ f共 0 兲 ⫹ ¯ ⫹
n!
1
⬁
⬁
兺
nb n Z
D n⫽1
n!
⫹
f 共 n 兲共 0 兲 Z n ⫹ ¯
(14)
with f (x) as the nth-order derivative of f(x). In an exactly analogous manner one obtains
1
g 共 m 兲共 0 兲 Z m ⫹ ¯
⫽ ␺ 0 共 0 兲 ⫹ ¯ ⫹ ␺ m 共 0 兲 Z m ⫹ ¯,
(15)
where
1
␺ m共 0 兲 ⫽
m!
g 共 m 兲共 0 兲 ,
with g ( m ) (x) as the mth-order derivative of g(x). Consequently, the scattering and absorption coefficients of the
medium layer can be expressed as
⬁
兺 ␾Z,
K ip 共 Z 兲 ⫽
l
l
(16a)
bn ⫽
D
兺
(16b)
l⫽0
B. Reflectance of a Media Layer
Generally speaking, the differential Eqs. (1) have no
known closed-form solutions, except in a few special
cases. Assuming, however, that any existing solutions
are analytic functions of Z, we can express them in series
form too, as
⬁
I⫽
兺
␺ l a m Z l⫹m .
n
D
n
冋兺
冋兺
共 ␾ l ⫹ ␺ l 兲 a n⫺l⫺1 ⫺
l⫽0
n⫺1
共 ␾ l ⫹ ␺ l 兲 b n⫺l⫺1 ⫹
l⫽0
兺
D n⫽1
⬁
na n Z
n⫺1
⫽
b mZ m.
(17)
⬁
兺兺
共 ␾ l ⫹ ␺ l 兲 a m Z l⫹m
l⫽0 m⫽0
⬁
⫺
⬁
兺兺
l⫽0 m⫽0
册
␺ l a n⫺l⫺1 .
l⫽0
(21)
These recurrence relations reveal two noteworthy features. First, there are only two undetermined coefficients, say a 0 and b 0 . All other coefficients, a n or bn for
n ⭓ 1, are functions of these. This is consistent with the
original 2 ⫻ 2 differential system. Second, both coefficients a n and bn are proportional to a n⫺1 /n and b n⫺1 /n,
which generally leads to sufficiently fast convergence of
the series expansions. Moreover, this last feature provides us with the possibility of truncating the series expansions at a certain finite order. Since the variable 0
⭐ Z ⬍ 1 is independent of the range of ink penetration,
Eqs. (20) and (21) allow us to draw meaningful conclusions.
Imposing the boundary condition at Z ⫽ 0 [Eq. (4)]
gives
(22)
The remaining undetermined coefficient, say a 0 , can be
determined by applying the boundary condition at the Z
⫽ 1 interface. Inserting Eq. (16) into the boundary at
Z ⫽ 1 [Eq. (5)], gives
⬁
兺
⬁
a n ⫽ I 0共 1 ⫺ r 0 兲 ⫹ r 1
n⫽0
兺
bn ,
n⫽1
⬁
I 0R ⫽ I 0r 0 ⫹ 共 1 ⫺ r 1 兲
兺
bn .
(23)
n⫽0
兺
These, together with Eqs. (16), can be inserted into the
differential equations to get the following algebraic system of equations:
⬁
兺
⬁
m⫽0
1
␺ l b n⫺l⫺1 , (20)
l⫽0
The reflectance of the system is then obtained as
a nZ n,
⬁
兺
兺
n⫺1
⫺
册
n⫺1
n⫽0
J⫽
(19)
b 0 ⫽ R ga 0 .
⬁
␺ lZ l.
兺兺
n⫺1
l⫽0
S ip 共 Z 兲 ⫽
⬁
By identifying the coefficients of Z n on both sides of the
equations one obtains, for n ⭓ 1,
an ⫽
f 共 n 兲共 0 兲 ,
m!
共 ␾ l ⫹ ␺ l 兲 b m Z l⫹m
l⫽0 m⫽0
(n)
g共 Z 兲 ⫽ g共 0 兲 ⫹ ¯ ⫹
兺兺
l⫽0 m⫽0
where
1
⫽⫺
⬁
⬁
⫽ ␾ 0 共 0 兲 ⫹ ¯ ⫹ ␾ n 共 0 兲 Z n ⫹ ¯,
␾ n共 0 兲 ⫽
n⫺1
1945
␺ l b m Z l⫹m ,
(18)
R ⫽ r 0 ⫹ 共 1 ⫺ r 0 兲共 1 ⫺ r 1 兲
bm
m⫽0
. (24)
⬁
兺
共 a m ⫺ b mr 1 兲
m⫽0
Since all the a n and b n are proportional to a 0 , the fraction in Eq. (24) is independent of a 0 . Thus one can compute the reflectance simply by setting a 0 to an arbitrary
value (say, a 0 ⫽ 1).
C. Transmittance of a Freely Suspended Media Layer
When the media layer is freely suspended as shown in
Fig. 2, the boundary condition at Z ⫽ 0 [Eq. (22)] should
be replaced by
1946
J. Opt. Soc. Am. A / Vol. 21, No. 10 / October 2004
Yang et al.
␰⫽
Fig. 2. Schematic diagram of a linear ink distribution in a freely
suspended layer. Ink penetration begins from the surface of the
substrate at z ⫽ D and ends at z ⫽ 0.
b 0 ⬇ r 2a 0 ,
where r 2 is the internal surface reflection at the lower interface. Even though the remaining expansion coefficients (a n , b n , n ⭓ 1) change with a 0 and b 0 , the expression for the reflectance at Z ⫽ 1 given in Eq. (24)
remains unchanged. It is important to note that the internal boundary reflection at the upper interface r 1 may
be different from that of the lower interface r 2 in the case
of an inhomogeneous ink distribution.
By combining the boundary conditions at Z ⫽ 1 [Eq.
(23)] with the condition at Z ⫽ 0,
I 0T ⫽ a 0共 1 ⫺ r 2 兲 ,
an expression for the transmittance of light through the
freely suspended layer T can be determined as
冒兺
⬁
T ⫽ a 0 共 1 ⫺ r 0 兲共 1 ⫺ r 2 兲
共 a m ⫺ b mr 1 兲.
(25)
m⫽0
Although a 0 explicitly appears in this expression, T is, interestingly enough, actually independent of a 0 , again because the a m and b m in the denominator are proportional
to a 0 as we have previously indicated.
D. Remarks
In the case of an inhomogeneous ink distribution, the expressions for S ip and K ip given in Eqs. (10) and (11) may
change form depending on whether the inequality
2h(S p , S i , Z) ⭐ h(K p , K i , Z) is satisfied. Suppose
the transition occurs at a certain depth ␰ ⭐ 1 determined
by
␳共 ␰ 兲 ⫽
␮ i ␣ i 共 2S p ⫺ K p 兲
␮ p ␣ p 共 K i ⫺ 2S i 兲
.
1 ␮ i ␣ i 共 2S p ⫺ K p 兲
␳ 1 ␮ p ␣ p 共 K i ⫺ 2S i 兲
.
Computed transition depths ␰ for the primary colors are
depicted in Fig. 3, which reveals a distinct dependence on
inks and spectra. This dependence is reasonable since
the transition occurs only in the absorption bands of these
colors. Moreover, the closer to the center of the absorption band, the lower the ␰. The vertical or nearly vertical
curves in the figure indicate that the ␰ value changes sign
in the vicinity of the wavelengths corresponding to a sign
change in K i ⫺ 2S i . For example, for cyan ink (solid
curve) and at ␭ ⬇ 430 nm, ␰ changes sign as a result of
decreasing absorption approaching the transparent band.
␰ again changes sign at ␭ ⬇ 540 nm approaching the absorption band of cyan.
After determining the transition depth ␰ for each color
and at each wavelength, the overall spectral reflectance of
the ink–paper composite can be calculated sublayer by
sublayer by repeatedly applying the theory developed
above. In each sublayer one needs only to redefine the z
coordinate as z ⫽ 0 and z ⫽ D corresponding to the bottom and top surfaces of a sublayer, respectively. For example, one can first compute the reflectance R 1 of the
lower sublayer. If we replace ␳ 1 with ␳ 1 ␰ , all equations
derived in Subsections 3.B. and 3.C. are immediately applicable. Then by considering all the medium below ␰ as
a backing for the upper sublayer with reflectance R 1 ,
namely, by replacing R g with R 1 in Eq. (22), we can compute the overall spectral reflectance of the entire ink–
paper mixture in exactly the same manner. These issues
are further illustrated in Subsections 4.B. and 4.C. below
with examples of linear and exponential ink distributions.
Similar operations apply even for transmittance.
Given that the transmittances of the lower and upper
sublayers are T 1 and T 2 , respectively, the overall transmittance of the entire ink–paper mixture is T ⫽ T 1 T 2 ,
provided the boundary reflection at the ␰ transition surface is negligible.
Clearly, such a stepwise strategy works even for a nonmonotonic ink distribution or, equivalently, for a
multilayer media system.
(26)
In such a case, one can divide the ink–paper layer into
two sublayers with transition surface at ␰. Within each
sublayer, S ip or K ip then has the same functional form.
Since ␰ 苸 (0, 1), a ␰ value outside this range implies no
transition.
For a monotonic ink distribution Eq. (26) has at most
one solution satisfying ␰ 苸 (0, 1). For a linear ink distribution, for example,
␳ 共 Z 兲 ⫽ ␳ 1 Z,
one obtains
(27)
Fig. 3. Spectral dependence of depths for status transition of
primary colors (linear ink distribution).
Yang et al.
Vol. 21, No. 10 / October 2004 / J. Opt. Soc. Am. A
4. EXAMPLES OF HOMOGENEOUS AND
INHOMOGENEOUS MEDIUM LAYERS
Equations (24) and (25) are general expressions giving
the reflectance and transmittance values for a medium of
arbitrary material distribution. Different distributions
lead to different recurrence relations Eqs. (20) and (21)
for the set of coefficients 兵 a n , b n 其 , which in turn describe
the different reflectance and transmittance properties.
To highlight the flexibility and utility of the solutions, we
shall apply the analysis to medium layers of both homogeneous and inhomogeneous ink distributions. In the
case of a homogeneous ink distribution, we focus on establishing connections between the present framework and
previous studies in the form of analytical solutions of the
K–M theory in both conventional and matrix
forms.2,4,13,14 As examples of inhomogeneous ink distribution, both linear and exponential ink distributions are
studied.
Since the matrix elements are constants, we can utilize
the definition of the exponential of a matrix to give
冋 册
⬁
兺
an
n⫽0
⬁
兺
⫽ exp
bn
⫽
␮p ␮p
x⫹
␳ 1 ␣ ip
␮i ␣i
y
is independent of Z. The coefficients (expanded in the vicinity of Z ⫽ 0) that apply in Eqs. (14) and (15) are then
K ⫽ ␾0 ⫽
再
S ⫽ ␺0 ⫽
再
otherwise
兺
an ,
兺
bn ,
R bulk ⫽
共 n ⭓ 1 兲.
,
,
and
Correspondingly, Eqs. (20) and (21) become simply
n
(28)
Eq. (28) is essentially the matrix expression for the solutions to the K–M differential equations (at z ⫽ D or Z
⫽ 1) introduced by Emmel and Hersch.13,15 By applying the boundary condition at Z ⫽ 0 [Eq. (4)], we can
compute the bulk reflectance of a homogeneous medium
(with backing R g ) by13,15
h 共 S p , S i , 1兲 ,
D
a0
.
w b0
J共 Z ⫽ 1 兲 ⫽
2h 共 S p , S i , 1兲 ⬎ h 共 K p , K i , 1兲
bn ⫽
v
u
⬁
2h 2 共 S p , S i , 1兲 / ␾ 0 ,
n
a0
b0
n⫽0
otherwise
an ⫽
t
I共 Z ⫽ 1 兲 ⫽
h 共 K p , K i , 1兲 ,
D
⫺ 共K ⫹ S兲
D
⬁
2h 共 S p , S i , 1兲 ⬎ h 共 K p , K i , 1兲
␺n ⫽ 0
S
册 冎冋 册
Considering that
关 2h 共 S p , S i , 1兲 h 共 K p , K i , 1兲兴 1/2,
␾ n ⫽ 0,
⫺S
n⫽0
Consequently, the function
h 共 x, y, Z 兲 ⫽
共K ⫹ S兲
冋 册冋 册
共 0 ⬍ Z ⭐ 1 兲.
1 ␣ ip
再冋
n⫽0
A. Homogeneous Ink Penetration
For the homogeneous ink penetration, one has
␳共 Z 兲 ⫽ ␳1 ,
1947
⫽
关共 K ⫹ S 兲 a n⫺1 ⫺ Sb n⫺1 兴 ,
v ⫹ R gw
t ⫹ R gu
共 R ⬁ ⫺ R g 兲 exp共 ⫺2bSD 兲 ⫺ R ⬁ 共 1 ⫺ R ⬁ R g 兲
R ⬁ 共 R ⬁ ⫺ R g 兲 exp共 ⫺2bSD 兲 ⫺ 共 1 ⫺ R ⬁ R g 兲
,
(29)
关 ⫺共 K ⫹ S 兲 b n⫺1 ⫹ Sa n⫺1 兴 .
where
It is convenient to rewrite these in matrix form, i.e.,
冋 册
冋
D 共K ⫹ S兲
an
⫽
bn
S
n
⫺S
册冋 册
a n⫺1
.
⫺ 共 K ⫹ S 兲 b n⫺1
By repeatedly applying the recurrence relation, one obtains
冋 册
冋
Dn 共K ⫹ S兲
an
⫽
bn
S
n!
⫺S
⫺ 共K ⫹ S兲
册冋 册
n
a0
.
b0
R ⬁ ⫽ 1 ⫹ K/S ⫺ 共 K 2 /S 2 ⫹ 2K/S 兲 1/2,
b ⫽ 共 1 ⫺ R ⬁ 2 兲 /2R ⬁ .
According to Eq. (24), the total reflection of the ink layer,
including the top surface reflection, can therefore be written as
1948
J. Opt. Soc. Am. A / Vol. 21, No. 10 / October 2004
R ⫽ r 0 ⫹ 共 1 ⫺ r 0 兲共 1 ⫺ r 1 兲
Yang et al.
R bulk
1 ⫺ r 1 R bulk
⫽ r 0 ⫹ 共 1 ⫺ r 0 兲共 1 ⫺ r 1 兲
⫻
共 R ⬁ ⫺ R g 兲 exp共 ⫺2bSD 兲 ⫺ R ⬁ 共 1 ⫺ R ⬁ R g 兲
共 R ⬁ ⫺ r 1 兲共 R ⬁ ⫺ R g 兲 exp共 ⫺2bSD 兲 ⫺ 共 1 ⫺ R ⬁ r 1 兲共 1 ⫺ R ⬁ R g 兲
.
␾ 1 ⫽ 关 h 共 K p , K i , 0兲 h ⬘ 共 S p , S i , 0兲
Equation (30) is exactly the analytical solution of the
K–M theory for a homogeneous material distribution.16
Similarly, Eq. (25) becomes
⫹ h ⬘ 共 K p , K i , 0兲 h 共 S p , S i , 0兲兴 / ␾ 0 ,
␺ 1 ⫽ 关 4h 共 S p , S i , 0兲 h ⬘ 共 S p , S i , 0兲
T ⫽ 共 1 ⫺ r 0 兲共 1 ⫺ r 1 兲
⫻
⫺ ␺ 0␾ 1兴 / ␾ 0 ,
共 1 ⫺ R ⬁ 2 兲 exp共 ⫺bSD 兲
共 1 ⫺ R ⬁ r 1 兲 2 ⫺ 共 R ⬁ ⫺ r 1 兲 2 exp共 ⫺2bSD 兲
.
and for n ⭓ 1 there are
Therefore, the conventional [Eqs. (30) and (31)] and matrix expressions of the K–M solutions for the homogeneous ink distribution derived previously are only special
cases of the present framework.
B. Linear Ink Penetration
To cope with the possibility that there exists a transition
for the inequality 2h(S p , S i , Z) ⭐ h(K p , K i , Z) at Z
⫽ ␰ (see Subsection 3.D.), the expression for a linear ink
distribution given in Eq. (27) is modified to
␳ 共 Z兲 ⫽ CZ ⫹ ␳ 0 ,
共0 ⬍ Z ⬍ ␰兲
共 Z ⫺ ␰ 兲/共 1 ⫺ ␰ 兲,
共␰ ⬍ Z ⬍ 1兲
.
Thus Z ⫽ 0 and 1 correspond to the bottom and top surfaces in each sublayer. In the sublayer below the transition surface (Z ⫽ ␰ ), ␳ 0 ⫽ 0 and C ⫽ ␳ 1 ␰ , while in the
sublayer above the transition surface, ␳ 0 ⫽ ␳ 1 ␰ and C
⫽ ␳ 1 (1 ⫺ ␰ ). Therefore
1 ␣ ip
␮p ␣p
x⫹
CZ ⫹ ␳ 0 ␣ ip
␮i
␣i
y.
For Z satisfying 2h(S p , S i , Z) ⭐ h(K p , K i , Z) the
Taylor expansion coefficients (expanded in the vicinity of
Z ⫽ 0) are
␾ 0 ⫽ h 共 K p , K i , 0兲 ,
␾ 1 ⫽ h ⬘ 共 K p , K i , 0兲 ⫽
␺ 1 ⫽ h ⬘ 共 S p , S i , 0兲 ⫽
␾ n ⫽ 0,
共 n ⬎ 1 兲,
␺ 0 ⫽ h 共 S p , S i , 0兲 ,
C ␣ ip
␮i ␣i
C ␣ ip
␮i ␣i
Ki ,
共 n ⬎ 1 兲,
while for Z satisfying 2h(S p , S i , Z) ⬎ h(K p , K i , Z),
according to Eqs. (A9)–(A12) (see Appendix A),
␾ 0 ⫽ 关 2h 共 K p , K i , 0兲 h 共 S p , S i , 0兲兴 1/2,
␺ 0 ⫽ 2h 2 共 S p , S i , 0兲 / ␾ 0 ,
m! 共 n ⫺ m 兲 !
m⫽0
n⫺1
兺
⫺
m⫽1
冋兺
n
␺n ⫽
␾ m ␾ n⫺m
2
␾0 ,
(33)
2h 共 m 兲 共 S p , S i , 0兲 h 共 n⫺m 兲 共 S p , S i , 0兲
m! 共 n ⫺ m 兲 !
m⫽0
n
⫺
册冒
兺
␾ m ␺ n⫺m
m⫽1
册冒
␾0 ,
(34)
h 共 x, y, 0兲 ⫽
h ⬘ 共 x, y, 0兲 ⫽
1 ␣ ip
␮p ␣p
C ␣ ip
␮i ␣i
h 共 n 兲 共 x, y, 0兲 ⫽ 0,
x⫹
C ␣ ip
␮i ␣i
y,
y,
共 n ⭓ 2 兲.
From Eqs. (33) and (34), coefficients ␾ n and ␺ n can be
computed from the lower-order coefficients. Consequently, the sets of expansion coefficients 兵 a n 其 and 兵 b n 其
and then the reflectance and transmittance values can be
computed by employing Eqs. (20) and (21) and Eqs. (24)
and (25), respectively.
C. Exponential Ink Penetration
Apart from a linear distribution, one can imagine an exponential ink distribution as another typical example of
inhomogeneous distribution. The driving force for this
kind of ink distribution is a chemical distribution gradient. In the exponential case
␳ 共 Z 兲 ⫽ ␳ 1 exp关 ␤ 共 z ⫺ D 兲兴 ⫽ ␳ 1 exp关 ␥ 共 Z ⫺ 1 兲兴 ,
Si ,
␺ n ⫽ 0,
h 共 m 兲 共 K p , K i , 0兲 h 共 n⫺m 兲 共 S p , S i , 0兲
where
Z/ ␰ ,
h 共 x, y, Z兲 ⫽
␾n ⫽
(32)
where Z is the rescaled coordinate variable
再
冋兺
n
(31)
Z⫽
(30)
0 ⭐ z ⭐ D,
(35)
where ␤ is a quantity characterizing the ink penetration
and D is an ‘‘effective’’ depth of ink penetration chosen
such that ␥ ⫽ ␤ D Ⰷ 1 (say, D ⫽ 5/␤ ). Consequently
exp(⫺␥) Ⰶ 1.
Similar to the treatment for linear ink distribution, to
cope with the situation when one must divide the layer of
Yang et al.
Vol. 21, No. 10 / October 2004 / J. Opt. Soc. Am. A
1949
ink penetration into two sublayers with the transition
surface at Z ⫽ ␰ , the expression given in Eq. (35) is modified into
␳ 共 Z兲 ⫽ C exp关 ␦ 共 Z ⫺ 1 兲兴 ,
0 ⭐ Z ⭐ 1,
where C ⫽ ␳ 1 exp关␥ (␰ ⫺ 1)兴, ␦ ⫽ ␥ ␰ ; and C ⫽ ␳ 1 , ␦
⫽ ␥ (1 ⫺ ␰ ) for the sublayers below and above the transition surface, respectively.
The derivatives of h(x, y, Z) at Z ⫽ 0 are then
h 共 x, y, 0兲 ⫽
1 ␣ ip
␮p ␣p
x ⫹ exp共 ⫺␦ 兲
h 共 n 兲 共 x, y, 0兲 ⫽ exp共 ⫺␦ 兲
␦ n C ␣ ip
␮i
␣i
C ␣ ip
␮i ␣i
y,
y,
(36)
共 n ⭓ 1 兲.
(37)
Therefore in the case 2h(S p , S i , Z) ⭐ h(K p , K i , Z),
the Taylor expansion coefficients (expanded in the vicinity
of Z ⫽ 0) become
␾0 ⫽
␺0 ⫽
1 ␣ ip
up ␣p
1 ␣ ip
up ␣p
C ␣ ip
K p ⫹ exp共 ⫺␦ 兲
S p ⫹ exp共 ⫺␦ 兲
␾ n ⫽ exp共 ⫺␦ 兲
␺ n ⫽ exp共 ⫺␦ 兲
␦ n C ␣ ip
n! u i ␣ i
␦ n C ␣ ip
n! u i ␣ i
ui ␣i
C ␣ ip
ui ␣i
Ki ,
Si ,
Ki ,
Si .
Conversely,
for
the
region
2h(S p , S i , Z)
⬎ h(K p , K i , Z), the expansion coefficients in the vicinity of Z ⫽ 0, viz., 兵 ␾ n 其 and 兵 ␺ n 其 , can be computed by replacing h ( n ) (x, y, 0) in Eqs. (33) and (34) with Eqs. (36)
and (37).
5. SIMULATIONS AND DISCUSSIONS
The simulation presented here is aimed at demonstrating
the methodology, flexibility, and applicability of the model
rather than at direct comparison with experiments. To
that end, two model systems of dyed sheets with homogeneous and linear ink distribution are studied. For ease
of comparison between these ink distribution models, we
assume that all of them have the same paper structure
and contain the same amount of dye for each color. The
K–M scattering and absorption powers of the paper S p z p ,
K p z p are shown in Fig. 9 of our previous work9 while
those of inks S i z i , K i z i are taken from Fig. 5 of same.
A. Homogeneous Media Distribution
As shown in Section 4, spectral reflectance values of a homogeneous medium can be computed either analytically
with Eq. (30) or numerically with the series solutions
given by Eq. (24). Comparison of the two approaches
with this example may allow us to test the convergence of
the series solutions. Figure 4 shows the predictions of
the spectral reflectance and transmittance values computed with different orders n of the series expansion.
Corresponding values computed analytically with Eqs.
Fig. 4. Convergence of the computed spectral reflectance values
of primary inks with respect to different order of series expansion. The corresponding values computed analytically with Eqs.
(30) and (31) are shown with dots (not visible under the n ⫽ 10
curve).
(30) and (31) are denoted with dots in the figures. It is
clear from the pictures that both the reflectance and
transmittance values are effectively converged when n
⫽ 10. In other words, the series solution demonstrates
fairly fast convergence.
To study the effects of ink penetration, comparative
computations with and without considering ink–paper
mixing were carried out. When there is no ink–paper
mixing, the ink forms a uniform ink layer on the paper
surface. To avoid any possible complications due to surface reflections, only bulk reflectance is compared. In
other words, quantities related to surface reflection (r 0 ,
r 1 , and r 2 ) in Eqs. (24) and (30) have been set at zero in
the computations. Figure 5 demonstrates distinct peak
and valley dependence on wavelength of illumination corresponding to transparent and absorption bands of inks,
respectively. The effects of ink–paper mixing may be described as peak depression and valley raising compared
with prints without ink–paper mixing. These effects occur because of the stronger scattering power of the ink–
1950
J. Opt. Soc. Am. A / Vol. 21, No. 10 / October 2004
paper mixture, resulting in a greater SIPV factor ( ␮ ip
Ⰷ ␮ i ) in the transparent bands. In other words the minor absorption powers of the inks are significantly amplified by an increase in scattering 关 K ip ⬇ ( ␮ ip / ␮ i )K i 兴 .
Also, because of the stronger scattering power, or the
presence of paper material with strong scattering power,
the ink–paper mixture becomes more reflective, which
leads to the valley raising in the absorption bands. The
combination of the depression and raising mechanisms
drives the ink–paper mixture to significantly lower color
saturation. This combination also has an effect on the
hue of the colors because of the nonlinear response of human color vision to light stimulation. These trends explain the well-known phenomena of hue shift and saturation reduction observed experimentally.
They also
explain the observed reduction of color gamut on ink
penetration.17
Yang et al.
Fig. 7. Ink distribution with respect to depth of ink penetration
z in cases of homogeneous and linear ink distribution.
B. Linear Media Distribution
The simulated spectral reflectance of an ink–paper mixture possessing a linear ink distribution (see Fig. 6) exhibits valley-raising features similar to those of the homogeneous ink distribution, even though peak depression
plays a more dominant role. This can be well understood
from the point of view of ink distribution. Since the ink
concentration decreases linearly from the upper surface
as shown in Fig. 7, the inks are concentrated mainly close
to the top surface. Therefore the increasing scattering
Fig. 8. Computed spectral reflectance of inked sheets with either homogeneous or linear ink distribution. The sheets have
the same thickness and contain the same total amounts of inks.
Fig. 5. Computed spectral reflectance of primary inks with and
without considering ink–paper mixing in the case of homogeneous ink distribution.
Fig. 6. Computed spectral reflectance of primary inks with and
without considering ink–paper mixing in the case of linear ink
distribution.
power in the ink–paper mixture (compared to the ink
layer) has a limited impact on the reflectance in the absorption bands. Nevertheless, it still plays a significant
role in the transparent bands.
It seems natural to make comparisons between the homogeneous and linear ink distributions. Since these two
systems contain the same amount of ink as well as the
same depth of ink penetration D, the differences exhibited
are indications only of the different ink distributions.
Figure 8 clearly shows the dependence of the computed
spectral reflective properties on the forms of the two ink
distributions. Generally speaking, the system with a homogeneous ink distribution exhibits larger reflectance
and transmittance than does the system with a linear distribution. The explanation for the larger reflectance
found with the homogeneous ink distribution is straightforward. Since both systems have the same amount of
ink, there is a greater ink concentration close to the top
surface in the case of a linear distribution (see Fig. 7)
through which the light must pass (if not already absorbed) before it is reflected by the paper material. Consequently, the dyed sheet with the homogeneous ink distribution is less absorptive (more reflective) than that
with the linear ink distribution. An important consequence of Fig. 8 is that it should be possible to distinguish
Yang et al.
homogeneous from inhomogeneous (e.g., linear) ink distribution by means of spectral reflectance measurements.
Tests carried out on the convergence of the series expansions show that the computed spectral reflectance and
transmittance values become essentially wholly converged at expansion order n ⫽ 10.
Vol. 21, No. 10 / October 2004 / J. Opt. Soc. Am. A
f 共 Z 兲 ⫽ K ip 共 Z 兲 ⫽ 关 2h 共 K p , K i , Z 兲 h 共 S p , S i , Z 兲兴 1/2,
(A1)
g 共 Z 兲 ⫽ S ip 共 Z 兲 ⫽ 2h 2 共 S p , S i , Z 兲 /f 共 Z 兲 ,
1 ␣ ip
h 共 x, y, Z 兲 ⫽
␮p ␣p
␳ 共 Z 兲 ␣ ip
x⫹
␮i
␣i
y.
For ease of application, Eqs. (A1) and (A2) are written as
f 2 共 Z 兲 ⫽ 2h 共 K p , K i , Z 兲 h 共 S p , S i , Z 兲 ,
f 共 Z 兲 g 共 Z 兲 ⫽ 2h 共 S p , S i , Z 兲 .
2
(A3)
(A4)
On the left-hand side of Eq. (A4) the nth order derivative of function f(Z)g(Z) equals
n
关 f 共 Z 兲 g 共 Z 兲兴 共 n 兲 ⫽
兺
m⫽0
n!
m! 共 n ⫺ m 兲 !
f 共 m 兲 共 Z 兲 g 共 n⫺m 兲 共 Z 兲
n
兺
⫽ f 共 Z 兲 g 共 n 兲共 Z 兲 ⫹
m⫽1
n!
m! 共 n ⫺ m 兲 !
⫻ 共 Z 兲 g 共 n⫺m 兲 共 Z 兲 .
f 共m兲
(A5)
On the right-hand side of Eq. (A4) there is
2 关 h 2 共 S p , S i , Z 兲兴 共 n 兲
n
⫽2
兺
m⫽0
n!
m! 共 n ⫺ m 兲 !
h 共 m 兲 共 S p , S i , Z 兲 h 共 n⫺m 兲 共 S p , S i , Z 兲 .
(A6)
Therefore
冋兺
n
g
共n兲
共Z兲 ⫽
m⫽0
2n!
m! 共 n ⫺ m 兲 !
h 共 m 兲 共 S p , S i , Z 兲 h 共 n⫺m 兲
n
⫻ 共Sp , Si , Z兲 ⫺
兺
m⫽1
⫻ f 共 m 兲 共 Z 兲 g 共 n⫺m 兲 共 Z 兲
n!
m! 共 n ⫺ m 兲 !
册冒
f共 Z 兲.
(A7)
In an exactly analogous manner, one obtains
冋兺
n
f 共 n 兲共 Z 兲 ⫽
m⫽0
n!
m! 共 n ⫺ m 兲 !
⫻ 共Sp , Si , Z兲 ⫺
h 共 m 兲 共 K p , K i , Z 兲 h 共 n⫺m 兲
1
n⫺1
兺
2
m⫽1
⫻ f 共 m 兲 共 Z 兲 f 共 n⫺m 兲 共 Z 兲
册冒
n!
m! 共 n ⫺ m 兲 !
f共 Z 兲.
(A8)
␺0 ⫽ g共 Z 兲,
(A9)
Considering that
␾0 ⫽ f共 Z 兲,
APPENDIX
According to Eqs. (10) and (11) and (12) and (13) the K–M
absorbing and scattering coefficients of the ink–paper
mixture are expressed as
(A2)
where
6. SUMMARY
Ink–paper interaction is a complex physical process depending on both the surface and bulk properties of paper
and the rheological properties of inks. The irregularity
and inhomogeneity of paper structure result in a variety
of forms of ink penetration. Experimental as well as theoretical studies of ink penetration have proven to be very
challenging because of the inherent complexity. Consequently, theoretical modeling becomes particularly important as it provides not only understanding and interpretation of measurements but also acts as a guide to
experimental practices.
In this paper we present a unified theoretical framework for studying light propagation in a layer of arbitrary
ink distribution. The framework is an extension of the
revised Kubelka–Munk theory taking into account major
effects of scattering on the path of light propagation. The
fluxes propagating downward and upward are represented by rapidly converging Taylor expansions. Existing formulas, the conventional and the matrix solutions of
the Kubelka–Munk theory, have been shown to be special
cases of the present framework. Applications to different
forms of ink distribution—homogeneous, linear, and
exponential—resulting from different mechanisms governing ink–paper interaction have been studied in detail.
Simulations of spectral reflectance consisting of peak (reflective) and valley (absorption) bands exhibit similar features of ink penetration for both homogeneous and linear
ink distributions. The effect of ink penetration is a combination of peak depressing and valley raising that results in significant hue shift, reduction of saturation, and,
in turn, reduction in color gamut.
An inhomogeneous ink distribution in printed paper
has been suspected of causing visible artifacts. However,
direct experimental characterization has proven to be
very difficult with no reliable measuring method yet
available to investigate the nature of the ink distribution.
Microtome cuts and confocal microscopy are but two
methods that have been applied. However, neither has
given reliable results. As been pointed out by one of the
reviewers, these ‘‘negative experiments’’ were never published. Considering the weaknesses of the original K–M
theory, it is our hope that theoretical developments like
the present study may be useful for experimental practices in the future. The present framework, still under
development, is a continuing effort aimed at characterizing ink penetration and testing, and it is a joint project of
research institutes and the paper-manufacturing industry.
1951
␾m ⫽
f 共m兲
m!
共 Z 兲,
␺n ⫽
g共n兲
n!
共 Z 兲,
(A10)
one gets
1952
J. Opt. Soc. Am. A / Vol. 21, No. 10 / October 2004
冋兺
n
␾n ⫽
m⫽0
1
m! 共 n ⫺ m 兲 !
n⫺1
1
兺
2
␾ 共 m 兲 ␾ 共 n⫺m 兲
m⫽1
冋兺
册冒
␾0 ,
(A11)
Corresponding author Li Yang’s e-mail address is
[email protected].
REFERENCES
1.
n
m⫽0
by the Swedish Foundation for Strategic Research
through the Surface Science and Printing Program
(S2P2).
h 共 m 兲 共 K p , K i , Z 兲 h 共 n⫺m 兲
⫻ 共Sp , Si , Z兲 ⫺
␺n ⫽
Yang et al.
2
m! 共 n ⫺ m 兲 !
h
共m兲
共 Sp , Si , Z 兲h
n
⫻ 共Sp , Si , Z兲 ⫺
兺
␾ 共 m 兲 ␺ 共 n⫺m 兲
m⫽1
共 n⫺m 兲
册冒
2.
3.
␾0 .
4.
(A12)
Therefore the nth order derivatives of functions f(Z) and
g(Z) and f ( n ) (Z) and g ( n ) (Z) can be computed from their
lower-order derivatives, as can those of h(x, y, Z).
The derivative of function h(x, y, Z) depends on the
form of ink penetration ␳ (Z). For linear ink penetration
and the derivatives are
h 共 1 兲 共 x, y, 0兲 ⫽
6.
7.
␳ 共 Z 兲 ⫽ CZ ⫹ ␳ 0
h 共 x, y, 0兲 ⫽
5.
8.
1 ␣ ip
␮p ␣p
C ␣ ip
␮i ␣i
h 共 n 兲 共 x, y, 0兲 ⫽ 0,
x⫹
␳ 0 ␣ ip
␮i ␣i
y,
9.
10.
y,
11.
共 n ⭓ 2 兲.
12.
For exponential ink distribution
␳ 共 Z 兲 ⫽ C exp关 ␦ 共 Z ⫺ 1 兲兴 ,
共 0 ⭐ Z ⭐ 1 兲,
13.
and the derivatives of h(x, y, Z) are then
h 共 x, y, 0兲 ⫽
1 ␣ ip
␮p ␣p
x ⫹ exp共 ⫺␦ 兲
h 共 n 兲 共 x, y, 0兲 ⫽ exp共 ⫺␦ 兲
␦ n C ␣ ip
␮i
␣i
C ␣ ip
␮i ␣i
y,
14.
15.
y,
共 n ⭓ 1 兲.
ACKNOWLEDGMENTS
The authors thank Patrick Emmel for valuable comments. Stimulating comments of the reviewers have
been particularly helpful. This work has been supported
16.
17.
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