Final Project’s Report :
Method for accurate and sensitive
Refractive Index Measurement
Advisor: Prof. S.G .Lipson
Moshe Tordjman
Physics Department -Technion
1
Contents
A- Experiment goals ----page 3
B- Theoretical Approach ----page 3
C- Experiment setup ----page 6
D- Simulations & Expected Results ----page 8
E- Experimental Results and Analysis ---- page 11
F- Conclusions and Interpretations ----page 27
G- Future Suggestions ---- page 27
H- Appendix ---- page 28
I- References ----page 31
2
A.
Experiment Goals :
We are interested to check the accuracy and sensitivity of an optical system due to minimal
changes of the order of ~ 10−6 in Index refraction in the near field.
The different indexes refractive we use for that were from gases such Alcohol and Helium.
Such a technique appears to be useful in many applications with additional developments
and improves like surface Plasmon resonance and many others.
B.
Theoretical Approach :
We use an optical system which detects the refractive index field with high spatial and
refractive index resolution.
So we follow the phenomena where the phase or amplitude of the reflected or refracted
waves are sensitive to the refractive index. A quantitative analysis of these phenomena can be
made by solving Maxwell’s equation with the corresponding boundary conditions at the
interface.
By using Fresnel’s equation , this equation describes the ratio between 3 waves:
- Incident (I).
- Transmitted (T).
- Reflected (R).
Which form on the surface separating materials with refractive indices μ1 , μ2 ( μ2 incident
refractive index, μ2 transmitted refractive index). We denote the amplitude of these waves
uuur
uur
ωμ1
by E EI , ER , ET and their wave vectors by K I , K R , KT where K I = K R = c and
uuur ωμ2
KT = c .
These three wave vectors lie in one plane, the incident plane, which is orthogonal to the
)
plane of the surface separating the media. We denote the incident angle by i and the
transmitted angle by r) and the angles between the wave vectors and the normal to the plane
surface n) . The reflection angle is equal to the incident angle. The general polarization of
these waves is defined by a superposition of two specific cases: when the vector EI is in the
plane of incidence, the polarization is denoted by mode P or and when EI is normal to the
plane of incidence it is denoted by S or ⊥ .
3
The Fresnel coefficients are defined by the complex ratios:
ER
RE=
ET
EI
,
TR=
EI
.
The coefficients of the S mode are:
)
)
)
)
μ cos i − μ2 cos r cos i − μr cos r
RE⊥ = 1
)
)
)=
)
μ1 cos i + μ2 cos r cos i + μr cos r
)
)
2μ1 cos i
2 cos i
TR⊥ =
)
)
)=
)
μ1 cos i + μ2 cos r cos i + μr cos r
The coefficients of the P mode are:
)
)
)
)
μ cos r − μ 2 cos i cos r − μ r cos i
RE// = 1
)=
)
)
)
μ1 cos r + μ 2 cos i cos r + μr cos i
)
)
2 μ1 cos r
2 cos r
TR// =
)=
)
)
)
μ1 cos r + μ2 cos i cos r + μ r cos i
Æ The phenomena of reflection with amplitude which is sensitive to the refractive index is :
Total internal refraction
Total internal refraction:
In the case of propagation of a wave through a surface separating different refractive
)
)
−1 μ
indices where μ2 < μ1 and i > sin ( 2 ) the value of r which obeys Snell’s law becomes a
μ1
complex number.
)
−1 μ 2
i
We define the angle c = sin ( ) as the critical angle, and investigate the behavior of the
μ1
reflection and transmission coefficients above this angle.
From Snell’s law we have:
)
) μ
sin r = 1 sin i ≡ (1 + β 2 )1/ 2 > 1
μ2
4
)
)
2
1/ 2
If we use the equality cos r = (1 − sin r ) = ±i β then the Fresnel coefficients:
)
cos i − i μr cos β
RE⊥ =
)
cos i + μr cos β
)
i β − μr cos i
RE// =
)
i β + μr cos i
And if we use again the equality
⎛ p⎞
p − iq
= exp[−2i tan −1 ⎜ ⎟ then we can have:
p + iq
⎝q⎠
⎛
⎛ μ β ⎞⎞
RE⊥ = exp ⎜ −2i tan −1 ⎜ r ) ⎟ ⎟
⎝ cos i ⎠ ⎠
⎝
)
⎛
⎛
⎞⎞
cos
i
μ
RE// = exp ⎜ −2i tan −1 ⎜ r
⎟⎟
β
⎝
⎠⎠
⎝
5
C. Experiment Setup :
The optical system in which we work consist of an experimental cell, constructed from a
closed box of Perspex optically attached to a glass (BK7) prism by an optical oil to the
hypotenuse of 90° .The cell was designed in purpose to flow into it different gases in contact
with the upper layer of the glass prism. It is illuminated by a collimated, filtered and coherent
beam from a He-Ne laser.
An image of the back plan of the glass prism tighten to the cell is obtained on a charge
coupled device (CCD) camera. All the experiment was applied through the corresponding
angle of total internal reflection.
The figure follow represent most of the optical tools of the system:
6
The experiment was made of two separates parts:
-1- Alcohol gas:
The main purpose was to verify the validity of the optical system for accurate index
refraction measurement. This was achieved after several optimizations and coordination of
all the optical tools by getting a fine imaging of the overall influence of the Alcohol when
it was flowing into the cell .
As well known the Alcohol have the properties to be heavier than Air and so his refractive
index is higher and correspond to n=1.00152 in front of n=1.0002761 for Air.
It can be mentioned that we succeed to observe the effect of Alcohol by naked eye before
we adventured ourselves to connect the system with the CCD.
-2- Helium gas:
Helium gas has the inverse properties, and so it is softer than Air and correspond to a
refractive index of n=1.000036 in front of n=1.0002761 for Air.
This fact represented a real challenge for the optical system which have for purpose this
time to detect the minimal influences of the entering Helium into the cell of Air and to get
an amplified imaging of this affecting process.
7
D. Simulations & Expected results:
Before getting to experimental approach, we tried do find out expected scale of influences
of different gases by taking into consideration the constraint of the digital monitor system to
be able to display as far as 4-5% of changes in any progressing process .
The Percentage Differences’ calculation Vs. Different Index Refraction where shown by
using the following gases with there refractive indexes respectively :
-Helium n=1.000036
-Air n=1.0002761
-Ethyl n=1.000871
-Alcohol n=1.00152
After remanipulating the Snell law and Reflection function from the theoretical approach
we can have :
R =
n 1 c o s (θ i ) − n 2 c o s (θ t )
n 1 c o s (θ i ) + n 2 c o s (θ t )
n 1 s i n (θ i ) = n 2 s i n (θ t )
θ t = a rc s in [
n 1 s i n (θ i )
]
n2
n 1 s i n (θ i )
])
n2
R =
n s i n (θ i )
n 1 c o s (θ i ) + n 2 c o s ( a r c s i n [ 1
])
n2
n 1 c o s (θ i ) − n 2 c o s ( a r c s i n [
Which we used in for computational simulation .
8
Reflect. vs. Incident.
1.1
Helium
Air
Ethyl
Alcohol
1
0.9
Reflection Coefficient
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
5
10
15
20
25
30
35
Incident Angle
40
45
50
55
60
Reflect. vs. Incident.
1.05
Helium
Air
Ethyl
Alcohol
Reflection Coefficient
1
0.95
0.9
0.85
0.8
41.5
41.55
41.6
41.65
41.7 41.75 41.8
Incident Angle
41.85
41.9
41.95
Reflection Coefficient vs. Incident angle for different gases
Correspond to Matlab code 1 in Appendix
9
42
Reflect. vs. Incident.
1.02
Reflection Coefficient
1
0.98
0.96
0.94
Helium
Air
Ethyl
Alcohol
0.92
0.9
41.8
41.82
41.84
41.86
Incident Angle
41.88
41.9
41.92
Reflection Coefficient vs. Incident angle for different gases after closing on critical points
Correspond to Matlab code 1 in Appendix
Than the corresponding reflection coefficient for the different critical angle can be deduced
as follow :
Gas:
Critical Points:
He
Air
Ethyl Alcohol
0.7298 0.7300 0.7305 0.7311
Critical Angles: 41.8122 41.8245 41.8550 41.8883
And so , close the critical angle we can be able to reach the sensitivity range of the optical
system imaging for the proposed gases affecting Air .
deg
Δ1 = Air − Alcohol = 0.0011 ⎯⎯→
0.0638°
deg
Δ 2 = Air − Ethyl = 0.0005 ⎯⎯→
0.0305°
deg
Δ 3 = Air − Helium = 0.0002 ⎯⎯→
0.0123°
10
E. Experimental Results & Analysis:
The analysis of our samples consist of substracting the images which were digitally
recorded from the CCD by using “differentiate” techniques of Matlab .
-1- For the first part of the experiment where Alcohol is inserted into the cell we had as
follow :
- The sample image reference without any Alcohol intervention as it has been imaged
[alchool3.tif]:
-The First effect with Alcohol intervention into Air reservoir , a clear black point appear as an
eventual influence of the Alcohol on the Reflection coefficient [alchool30.tif] :
11
- Second effect with Alcohol intervention into Air reservoir [alchool50.tif]:
- Third effect with Alcohol intervention into Air reservoir [Alchool70.tif]
12
- Differentiation result between Alchool3.tif & Alchool30.tif, the colored point show us the
“translated” appearance effect of Alcohol by the “differentiate” techniques of Matlab :
- Here is the same differentiation result by inversing images differentiation Alchool30.tif &
Alchool3.tif:
13
- Differentiation result between Alchool3.tif & Alchool50.tif and Alchool50.tif &
Alchool3.tif :
14
- Differentiation result between Alchool3.tif & Alchool70.tif and Alchool70.tif &
Alchool3.tif :
15
-2- For the second part of the experiment where Helium is inserted into the cell we had
as follow :
Since the capture of the sample was continuous (by video film) because the effect to be very
fine and brief. Here we should compare the images to the difference between two successive
images taken with no helium entry, to be sure that the result is not just the result of drift. We
recorded 3 different successful effects of it as follow :
Æ The 1st Helium effect :
- The reference frame without any Helium intervention in the Air reservoir
[helium32.06.tif]:
- The next frame show a starting of Helium intervention in the Air reservoir
[helium32.07.tif] :
16
- The last frame with Full Helium intervention into the Air reservoir, it can be notice the
progressing appearance of what look like a “ spreading “ which express the influence of
Helium presence on the back plan of the glass prism into the cell [helium32.08.tif] :
- Differentiation analysis results of the progressing intervention of Helium into Air reservoir
Helium32.08.tif - Helium32.07.tif, it can be noticed that the effect is clearly determined by
using the full range of colors in the display.
- With no Helium intervention:
17
- With Starting Helium intervention :
-With Full Helium intervention :
18
It can be show all over together for concluding the 1st Helium effect:
Æ The 2nd Helium effect recorded :
- The reference frame without any Helium intervention in the Air reservoir
[helium32.12.tif]:
19
- The next frame with a starting Helium intervention in the Air reservoir [helium32.13.tif]:
- The last frame with Full Helium intervention in the Air reservoir [helium32.14.tif]:
20
- The corresponding differentiation analysis results of the 2nd progressing intervention of
Helium into Air reservoir Helium32.13.tif - Helium32.14.tif.Here again, it can be noticed that
the effect is clearly determined by using the full range of colors in the display:
- With no helium :
- With Starting Helium intervention :
21
- With Full Helium intervention :
And putting all over the 2nd effect together:
22
Æ The 3rd Helium effect :
- This 3rd effect differ from other 2 first effect in the fact that there is 3 progressing states
before Helium intervention, during Helium intervention and after Helium intervention (this
last state hasn’t be checked in the other effects)
- Before Helium intervention [helium32.24.tif]:
- During Helium intervention [helium32.25.tif]:
23
- After Helium intervention [helium32.26.tif]:
- Then , the corresponding Differentiation analysis results of the 3rd progressing “ending”
intervention a of Helium into/from Air reservoir Helium32.25.tif - Helium32.24.tif.It can be
noticed that this effect is especially accented relatively to the 2 first others effects and so the
influence of Helium here is perfectly demonstrated by using the full range of colors in the
display:
- Helium intervention :
24
-Remaining Helium from the intervention:
- After all the Helium intervention:
25
Then, putting all over the 3rd effect together it gives:
All the results for “differentiation” techniques of Matlab are found in Appendix under Matlab
code 2 .
26
F. Conclusions & interpretations:
-
-
-
-
In our experiment we describe optional methods for measuring refractive index with
high lateral and refractive index resolution .
We inspected the requirements for imaging with high lateral resolution and high
sensitivity to the refracting index resolution.
We show that in order to achieve such sensitivity we have to use optical phenomena
which are sensitive to the refractive index, such as total reflection. In this phenomena
the phase/amplitude/polarization of the refracted or reflected wave don’t depend only
on the refractive index but also on the incident angle and polarization direction ,
therefore in order to get a sensitive refractive index measurement by using these
phenomena it is necessary to fix the polarization direction and the incident angle, or in
order to achieve high lateral resolution , illuminating the sample and collecting the
reflected wave should be done from all directions and at large incident and reflected
angles those facts bring us to : Total Internal Reflection .
In this method we create the required illumination by a combination of objective
lenses , where the objectives are used both for illuminating the sample and for imaging
the reflected waves .
We described 2 different gases (Alcohol & Helium) for evaluate the sensitivity of the
optical system through the imaging method based on measuring the change in the
amplitude and than had been analyzed by measuring the changing process in the
amplitude of the sample.
For Alcohol gas intervention we had a big and clear effect ( even for a naked eye) in
front of the Helium gas intervention which were very hard to be captured.
It has been shown from the experimental results for the Helium gas intervention, that
we were able to recompose the fine progressive change in the amplitude changes. We
could however see along this progress an organized recombination of what look like
fringes , the more the presence of Helium the more this kind of fringes were more
ordered.
G. Future suggestions:
- It is immediately required ( in my humble opinion) to figure out a “perfect” sensitive
imaging with the combination of a glass prism made of NIM ( Negative Index
Material); but such an application should come with more rigorous arguments.
( which I hope to have the opportunity to be involve into )
27
H. Appendix:
I. References :
-
S.G. Lipson, H. lipson and D.S Tannhauser , Optical physics.
W. Smith , Modern optical engineering.
E.Raz S.G. Lipson , E. Polturak , phys Rev.A40 1088(1989)
S.Kostianovsky , S.G Lipson , E.Ribak, appl opt. 32,4744(1993)
A.G Notcovich, I braslavsky, S.G Lipson ,J.Cryst.Growth 98,10(1999)
A.G Notcovich, V.Zhuk , S.G Lipson .App.Phys .76,13(2000)
28