Figure 1 is a cross section view of a substrate having some arbitrary surface profile.
Between the substrate and the glass window is a fluid that contains fluorescent dye. The
goal is to take an image (from the camera point of view in figure 1) in which the intensity
value in each pixel of the camera can be directly correlated to a specific value for fluid
layer thickness. The amount of light collected by each camera pixel is dependent up
excitation (or incident light) intensity. Even with the highly columnated light of a laser,
there will still be intensity variations across an image. To cancel out the affects of
incident light variations, we can use 2 fluorphores that respond to the incident
illumination in the same way. If we collect the light from each fluorophore in 2 separate
cameras then divide the 2 images, we should attain an image independent of the incident
light and only dependent upon the amount of fluorophore present, which corresponds to
fluid film thickness. This process is called Dual Emission Laser Induced Fluorescence
(DELIF)
Camera
Glass Window
Substrate
Figure 1. Typical DELIF Experiment.
Copetta and Rogers (Experiments in Fluids, 1998) and Hidrovo and Hart (Measurement
Science and Technology, 2001) have modeled a DELIF system to measure thin film pH
and measure the profile of a US quarter, respectively. These systems use 2 dyes
dissolved in the fluid layer as the 2 fluorophores. A 1-camera pixel sized subsection of
this system is represented in figure 2. When taking the ratio of the 2 dye intensities If2/If1,
in this system there is a linear relationship between the ratio and fluid layer thickness.
Figure 2. Standard DELIF geometry for 2 dye system.
Here, I will discuss a slightly different DELIF system, illustrated in figure 3, in which the
substrate is a fluorophore and there is only 1 dye dissolved in the fluid layer. The
motivation for modeling this system is Chemical Mechanical Planarization (CMP), a
process which is often used in the silicon industry to manufacture integrated circuits.
Silicon wafers are planarized by rubbing a polishing pad and chemically reactive slurry
against the silicon surface. To study the fluid dynamics of the CMP system we would
like to use DELIF to track the thin film slurry layer thickness during polish. Many CMP
polishing pads are made of polyurethane which fluoresces upon excitation by UV light.
The polishing slurry often contains nanoparticles abrasives, which strongly scatter light.
Because our DELIF system contains scattering particles and a fluorophore that only emits
light at x=t (as opposed to both florphores emitting throughout the fluid layer), we need
re-examine this system to verify that DELIF will produce an image we can accurately
calibrate to fluid layer thickness.
Figure 3. Model system geometry. The high energy wavelength fluorophore is the pad
(substrate). The reabsorbing fluorophore is the dye dissolved in the fluid. The fluid also
contains large colloidal particles that scatter light (we will assume isotropic scattering). In
the following section pad=substrate
3. System Model
3a. Emission Intensities
If light spreading is isotropic, the spreading function should be the same in all direction
and therefore we can consider the light along a unidirectional path. Lambert’s Law gives
the excitation intensity as a function of depth into the fluid due to the laser, Ie(x):
I e ( x)  I 0 e  ( l )Cx
(1)
Where (l) is the extinction coefficient, C is concentration of light absorbing/scattering
particles, and I0 is the laser intensity at x=0. The extinction coefficient is the sum of the
absorption coefficient, , and the scattering coefficient, , for the entire fluid layer,
      ( dye   p )   p
(2)
Where the subscript, p, stands for the scattering particles and the subscript, dye, stands for
dye particles. Scattering is dominated by the scattering particles and not the dye, p >>
dye, therefore dye is not included in equation 2. Similar to equation 2,
C  C p  C dye
(3).
We can solve for If,sub and If,dye by considering the radiative transfer equation (5):
(4)
dI   em dx   ext Idx
Where ext and em are the emission and extinction coefficients, respectively. Note that
the in the absence of the emission term, equation 4 is the differential form of Lambert’s
Law.
Considering a differential length, dx, through the fluid, we can calculate the intensity of
the substrate fluorescence as a function of depth, If,sub(x), into the fluid by considering the
following differential equation assuming uniform distribution of dye and particles.
dI f ,sub  I f ,sub[ dye ( )Cdye   p ( )C p ]dx
(5)
The extinction term here is positive because extinction of the substrate emission only
happens in the –x direction. The emission is absorbed by the dye in the fluid layer and
scattered and absorbed by the particles. There is no emission term in equation 5 because
the substrate only emits at x = xL, which is expressed in the boundary condition:
(6)
I f ,sub ( xL )  I e ( xL ) sub (l )subsub ( )  I 0 sub (l )sub sub ( )e  (l )CxL
Where sub(l) is the absorption of the substrate at the wavelength of the laser, sub, is the
quantum efficiency of the substrate, sub() is the emission efficiency of the substrate,
and L is the total distance from the fluid surface to the substrate. Equation 5 does not
account for backscattering of reflected laser light due to the particles in the fluid and.
assumes that all light is emitted back in the –x direction.
Integrating equation 5 yields the substrate fluorescence as a function of position, x, in the
fluid.
I f , sub ( t ) dI f ,sub
t

(7a)
I f ,sub( x ) I f ,sub x [ dye ( )Cdye   p ( )C p ]dx
I f ,sub ( x)  I 0 sub (l )sub sub ( ) exp([  dye ( )Cdye   p ( )C p ]( x  xL )   (l )CxL ) (7b)
The expression for the fluorescence intensity of the dye in the fluid layer as a function of
depth is found by solving the differential in equation 8:
dI f ,dye  [ I e ( x) dye (l )  I f ,sub ( x) dye ( )]Cdyedye dye ( )dx  I f ,dyeC p p ( )dx
(8)
The dye has two excitation sources represented in the first term of equation 8, the laser
and the substrate emission. Assuming that the dye does not reabsorb its own emission,
the only extinction source is the scattering and absorption due to the particles, which is
represented in the second term of equation 8. The boundary condition for the differential
equation 8 is that at the infinitely small boundary between the fluid layer and the
substrate at x = t, there is no scattering of the fluorescent light from the dye:
I f ,dye (t )  [ I e (t ) dye (l )  I f ,sub (t ) dye ( )]dyeCdye dye ( )
(9)
I f ,dye (t )  [ dye (l )   dye ( ) subsub sub ( )]I 0dyeCdye dye ( )e  ( l ) Ct
The solution to equation 8 applying the boundary condition in equation 9 is (quite a bit
more complicated):


 e  ( l )Cx  (C p p ( )  C (l )  1)e  ( l )Ct e  p (  )C p ( xt ) 

 (l )



C p p ( )   (l )C




I f ,dye ( x)  I 0Cdyedye dye 

(  (  ) C  2 (  ) C )( x t )


I f ,sub ( x)  1  (2C p p ( )  Cdye dye ( )  1)e dye dye p p

 dye ( )


I
2
C

(

)


(

)
C
0
p
p
dye
dye



(10)
Equations 1, 6 and 10 are graphed below (normalized) showing their relative intensity
through the fluid layer.
Intensity profiles for system fluorophores
100%
90%
80%
60%
Substrate
Incident Laser
light
% Highest Intensity
70%
50%
40%
30%
20%
Isub(x)
Idye(x)
Ie(x)
10%
0%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
% Depth in Fluid Layer
The cameras will observe If,dye(0) and If,sub(0) across an area with varying I0. In order to
attain thicknesses, t, that corresponds with image intensity we must cancel out the affect
of the incident light variation. Since I0 is contained in both the expression for the
substrate and the dye, taking the ratio of the 2 fluorphores will yield an image that can be
calibrated to fluid layer thickness.
 (  )C t



e p p  C p p ( )  C (l )  1

 ( )e dye (  ) Cdyet 

dye
l
  sub (l )sub sub ( )(C p p ( )  C (l )) 

I f ,dye (0)
R(t ) 
 Cdyedye dye ( ) 

 (  ) C 2 (  ) C t
I f ,sub (0)
  1  (2C p p ( )  Cdye dye ( )  1)e dye dye p p 

 ( )
 
dye

2C p p ( )  Cdye dye ( )

 

(11)
For thin films (small t), this equation takes the form of a quadratic (using exp(u)=u+1):
(12)
R(t )  At 2  Bt  D
Where A, B and D are constants that can be determined by calibration.
A
Cdyedyedye ( ) dye (l )Cdye dye ( )C p p ( )
 sub (l )sub sub ( )[C p p ( )  C (l )]
(13a)
  dye (l )C p p ( )[1  Cdye dye ( )]

B  Cdyedye dye ( ) 
  dye ( )[ 2C p p ( )  Cdye dye ( )  1]
 sub (l )sub sub ( )[C p p ( )  C (l )]

(13b)


 dye (l )[C p p ( )  C (l )]
D  Cdyedye dye ( ) 
  dye ( )
(13c)
 sub (l )sub sub ( )[C p p ( )  C (l )]

Traditionally in DELIF, R(t) is linear for thin films. However, in this scenario, the
scattering particles create the quadratic term. If Cp0, A0 and therefore, R(t) becomes
linear (greatly simplifying calibration of DELIF).
Questions to answer:
1. What values of t and Cp allow for DELIF calibration in the linear regime?
2. Can we calibrate the quadratic relationship?