Refractive Index Measurement Using the Michelson Interferometer
Content
Optical Fiber Communications Laboratory Report No. 4
REFRACTIVE INDEX MEASUREMENT USING THE MICHELSON INTERFEROMETER
Sahand Noorizadeh Charlie Hunter ECE 4501 – Georgia Institute of Technology NOVEMBER 2009
INTRODUCTION:
Refractive index is a measure of how much the speed of light (or other forms of waves) is reduced inside a medium compared to the speed of light in a vacuum. It is denoted by n and defined by Eq. (1)
n= c V
(1)
where c is the speed of light in vacuum and V is the speed of light in the medium. For example, the refractive index of light in air is about 1.0003 which means that the speed of light in air is very close to c . When light hits a boundary of two different indices of refraction, a portion of the incident wave is reflected while the rest is propagated through. The angle of propagation of the transmitted wave changes (bends) due to the change in refractive index. This behavior of light in materials with different refractive indices that makes waveguiding possible in optical fibers. Light propagates more slowly through media with higher refractive indices. Because of this, the optical path of light is equal to the physical distance times the refractive index of the medium, and therefore an optical distance is not the same as the physical distance. This property can be used in conjunction with a Michelson interferometer to measure the refractive index of a material with a known thickness. The Michelson interferometer was invented in early 1880's for the famous “failed” Michelson-Morely experiment to detect aether flow. In a modern Michelson interferometer, a coherent light beam, such as a laser, is split into two different beams and then reflected back to construct a single beam again. The newly joined beam forms an interference pattern with minima and maxima (fringes) that are exactly one wavelength apart. Introducing a material with a higher refractive index than air to the path of one of the split beams causes a change in the optical path. Observing the movement of these fringes as the optical path changes can give insight to the refractive index of that material. This lab report is concerned with setting up and using a Michelson interferometer to measure refractive indices of two glasses with known thickness. 1
THEORY:
In constructing a model to obtain the refractive index of a material with known thickness, two basic principles of optics were used and are briefly described in Appendix B.
THE MICHELSON INTERFEROMETER:
In 1887, before the discovery of light as a form of wave, Albert Michelson and Edward Morley conducted an experiment using a set-up similar to the one shown in Figure 1 to search for motion through aether. After not obtaining any result close to their expectation, the experiment was declared a failure. It was concluded later that their experiment had shown that the hypothesis of a stationary, luminiferous ether was not correct.
As shown in Figure 1, the Michelson interferometer uses a beam splitter to split a coherent light into two different beams and sends them in different directions. A pair of perpendicular mirrors
M irrors
Translation Stage
LASER Source M irrors
Beam Splitter
Screen
Optical Fringes
Figure 1. The Michelson interferometer. 2
mounted on translation stages are used to return the beams to the beam splitter at the same point to reconstruct the initial single beam. If the returning beams perfectly line up, an interference pattern will be formed in the reconstructed beam with minima and maxima as indicated by optical fringes observed on a screen. The gap between each fringe is exactly one wavelength (of the source beam). Moving any mirror pair on any of the interferometer's arms along their incident beams would cause the fringes to move and if they are displaced by a distance equal to the distance of two neighboring fringes, it would mean that the mirror pair was displaced by one wavelength.
APPARATUS
The objective of this lab was to measure refractive indices of a red glass and a microscope slide using the Michelson interferometer with a 633 nm He-Ne laser source. The apparatus shown in Figure 3 was assembled to achieve this objective. The first step of this experiment was to set up the Michelson interferometer of Figure 1 and adjust the position of the two mirror pairs to line up the returning split beams and observe the optical fringes on a screen. The next step was introducing a glass mounted on a rotary translation stage to the beam path of one of the interferometer's arms.
The idea was to use the fact that the overall OPL of the light would increase when it passes through the glass (that has a higher refractive index than air) and then rotate the glass to increase the distance that the light had to travel through the glass which would in turn increase the OPL and move the fringes. The increase in the traveling distance of a light beam through a material by rotating the material is shown in Figure 2. The rotary translation stage used in this experiment had a resolution of 1°.
Since counting the number of fringe shifts by eyesight was very difficult, the reconstructed beam was magnified and projected onto a photo multiplier with a pinhole on its aperture to detect the 3
minima and the maxima as the fringes moved. An adjustable speed chopper was placed on the path of the beam to periodically intersect the beam in order to make “the absence of the beam” as the oscilloscope reference point. The set-up of this experiment is shown in Figure 3.
l0
l1
l2
l3
1
2
3
Figure 2. Increase in the traveling distance of a light beam through a material by rotating the material. Mirrors
Glass Under Test 630nm He-Ne LASER
Rotary Translation Stage
Mirrors Beam Splitter Magnifier Oscilloscope
Adjustable Speed Chopper
Photo Multiplier Figure 3. The apparatus used to measure the refractive index of a glass with known thickness using the Michelson interferometer. 4
The job of the photomultiplier was to detect the incoming photons and generate an electric voltage proportional to the number of photons. This voltage was monitored with an oscilloscope. The minama (dark spots) of the interference pattern generated no voltage (reference level) and the maxima (bright spots) generated the highest negative voltage. A change of voltage from maximum negative to the reference level back to the maximum negative indicated one fringe shift. The speed (frequency) of the chopper had to be adjusted to obtain a periodic signal with minimum variation on its peaks.
DATA:
First, the red glass was mounted on the rotary stage and was set to the normal incident angle which was found by matching the beam reflection spots from the back of the red glass to the visible beam spots on the corresponding mirror pair. The angle of the rotary translation stage at this position was used as the reference angle (0°). The data collected for the red glass are listed in Table 1. The red glass intersected both the incoming beam from the beam splitter and the returning beam from the mirror pair. Table 1 – Rotation Angles of the Red Glass and the Corresponding Number of Fringe Shifts
Angle (Degrees) # of Fringe Shifts
0 0
2 4
4 9
6 14
8 27
10 47
12 68
14 97
16 128
18 166
The same process was repeated for a clear microscope slide but for this part of the experiment, only the returning beam from the mirror pair was intersected by the microscope slide. The data collected for the microscope slide are listed in Table 2.
Table 2 – Rotation Angles of the Microscope Slide and the Corresponding Number of Fringe Shifts
Angle (Degrees) # of Fringe Shifts
0 0
4 2
8 7
12 17
16 33
20 50
24 73
28 97
32 126
36 163
5
The thicknesses of the two glasses were measured with a micrometer and are listed in Table 3. Table 3 – Thickness Measurements
Material Red Glass Microscope Slide Thickness – mm 3.12 1.34
CALCULATIONS:
The last step in finding the refractive index of the two materials tested in this experiment was developing a theoretical model that could relate the number of fringe shifts to the rotation angle, thickness of the material, and of course the index of refraction. A model was developed based on the concept of Optical Path Length and its derivation is presented in detail in Appendix A. Eq. (2) and Eq (3) were developed for the red glass and the microscope slide respectively to model the number of fringe shifts as a function refractive index of the material n 2 , laser source wavelength 0 , rotation angle , and the thickness of the material d . m= d 2 1 1 1− 0 n2
2
(2)
d 1 m= 1 1− 2 0 n2
(3)
Eq. (2) is missing a factor of 1/2 because the red glass intersected both incoming beam from the beam splitter and the returning beam from the mirror pair which doubled the OPL of interest for the calculations.
To find the refractive index, a MATLAB® script was written to do the curve fitting routine by performing the following steps: 1. Generating an array of n 2 ranging from 1.000 to 1.900 with increments of 0.001. 6
2. Calculating m given by Eq. (2) and Eq. (3) for each value of n 2 for all angles of rotation. 3. Iterating through all sets of m generated in step 2 and comparing each one of them with the measured set of m and calculating the statistical error variance given by Eq. (4).
1 error= F n2 = N
i= N i=0
∑ mn2 meas.−mn 2 generated 2
(4)
4. Finding the value of n 2 that resulted in the least error variance.
Figure 4 and Figure 5 show the best fit curve to the measured data points of the red glass and microscope slide respectively. For the microscope slide, only data from rotation angles of less than 20 degrees were used to find the refractive index because in deriving the model, small angle approximation were used in applying the Snell's law.
R 1 8 0 n 2 1 6 0 1 4 0 1 2 0 N u m b e r o f F rin g e s 1 0 0 8 0 6 0 4 0 2 0 0 M B 2 4 6 A 8 1 0 n g le - D e a s u r e d e s t F it C 1 6 1 8 a t a u r v e D = 1 . 4 9 e d G la s s C o u n t e d N u m b e r o f F r in g e s v s .
A
0
1 2 1 4 e g r e e s
Figure 6. Best fit curve to the data obtained for the red glass. 7
M 6 0 n 2 5 0
ic r o s c o p e = 1 . 6 3
S
li d e
-
C
o u n t e d
N
u m
b e r
o f
F
r i n g
4 0 N u m b e r o f F rin g e s
3 0
2 0
1 0 M B 0 0 2 4 6 8 A 1 0 n g le e a s u r e d e s t F it C 1 8 2 0 a t a u r v e D
1 2 1 4 1 6 - D e g r e e s
Figure 7. Best fit curve to the data obtained for the microscope slide.
Table 4 lists the refractive indices of these two materials. The MATLAB® script used to generate these results is available in Appendix C.
Table 4 – Indices of Refraction Found by the Best Fit Curve Method
Material Red Glass Microscope Slide Index of Refraction 1.49 1.63
8
DISCUSSION:
Expected Results: Two different indices were obtained for each glass sample because two different
models were used: one that took into account the change in trajectory due to refraction and one that assumed a straight path through the material under test. When using the refraction model, the physical distance through the glass sample was less than the straight path model. The optical path length did not change between models ( the same counted fringes data was used) and optical path length is equal to the refractive index times the physical distance. Therefore, the refractive index was expected to be lower in the straight path model compared to the refraction model because the physical distance was smaller. For the microscope slide, it was expected that the refractive index would be close to 1.44 for the refraction model, which is the value for standard glass, and less than 1.44 for the straight path model. For the red glass, it was expected that the refractive index would be greater than 1.44 because of the dopants used in the glass to create the filter. So the refraction model was expected to yield a result very close to the actual refractive index, and the straight path model to yield a result less than the refractive index. Experimental Results: The refractive indices for the microscope slide were 1.37 for the straight
path model and 1.62 for the refraction model. The result from the refraction model was much larger than expected. The refractive index was less for the straight path model, and there was a significant difference between the two models. It is likely that the microscope glass has properties different than standard clear glass, perhaps added dopants, that increase the refractive index. The red glass refractive indices were 1.33 for the straight path model and 1.49 for the refractive model, very close to standard glass. Both of these values were less than expected and neither were greater than the clear glass microscope slide values. Again, the difference in the models was significant, with the refraction model resulting in the larger value. Dopants were used in the red glass 9
to create the filter effect, which would increase the refractive index. However, that filtering effect is dependent on wavelength. Because a red laser was used, it is likely that the wavelength of the source was in the pass band of the filter, causing the filter glass to act like standard clear glass: having a refractive index near 1.44. If an error occurred, it could have been one of two things: an error in the measurement, or an error in the model. The model worked according to expectation because the values of the refraction model were larger than the values of the straight path model, and all of the values were between 1.3 and 1.7. There were several places where an error could have occurred in the measurement process. Because the small angle approximation was used, all the data for angles greater than 20 degrees was ignored. For the microscope slide, that left the data to just 5 points, which may not have been enough to result in an accurate refractive index. Also, it is very difficult to create an exact 90 degree angle with the mirrors in the arms of the interferometer. When the refraction of light through the glass moved the output beam, it would return with a slightly different optical path if the mirrors were not precisely 90 degrees apart. This could have caused an extra fringe to be counted if the path was off by just one wavelength (633 nanometers). It is also possible that the measured thicknesses of the glass samples were not uniform throughout the sample, causing the thickness measurement to be inaccurate. Again, one wavelength difference would result in one fringe difference.
CONCLUSION:
An interferometer was used to measure the refractive index of two sample glass materials: a clear microscope slide and a red filter glass. This was done by measuring the fringe shifts of the beam output when the sample glass was rotated in the path of one of the interferometer arms. The shift is change in optical path, which is dependent on the refractive index of the material. Curve fitting models were developed and used to find the refractive index based on the data taken. For the microscope slide, 10
the results were higher than expected. For the red filter glass, the result were lower than expected. However, exploring the possibilities of what caused these results gave insight as to what could have happened. The microscope slide may have been made of a different material than originally assumed. The red glass filter most likely had a low absorption band at the wavelength of the source.
11
Appendix A
Derivation of the Theoretical Model for Calculating the Number of Fringe Shifts Based on Change in the Optical Path
d
n1 n2 2 1 2 n1 1 l3
l1
l2
d1
L
Figure A.1. Refraction of light ray inside a rotated glass.
From Figure A.1, the overall physical length of the light ray: L=l 1 d 1l 3 → l 3=L−l 1−d 1 l 2= d cos 1 and d 1=l 2 cos 1−2= d cos 1− 2 cos 2
and the optical path of the ray is:
OP=n1 l 1 n 2 l 2n1 l 3
=n 1 l 1n2
d n L−l 1−d 1 cos 1 1
=n 2
d [ cos 1 cos 2 sin 1 sin 2 ] d n 1 L−n 1 cos 1 cos 2 d n L−l 1−d 1 cos 1 1
=n 1 l 1n2
=n 2 d
tan2 n L−n1 d cos 2 −n 1 d sin 1 tan 2 sin 2 1
→ OP=n1 d tan 2 Using:
[
n2
2 1
1 −sin 1 n1 L−n1 d cos 1 sin 1 n
2
]
n 1 sin 1 =n 2 sin 2
Snell's law of refraction. n1 sin1 n2
tan 2=
sin 2 sin2 = = cos 2 1−cos 2
1−
n1
2
Trigonometry property of tan .
n2 2
sin 1
2
OP Can be written as: n1 sin 1 2 n2 n1 1 OP= n1 L−n1 d cos 1 2 2 sin 1 −sin 1 n2 n 1− 1 sin 2 1 n2 2 n1 d
[
]
The change in the optical path when the glass is rotated by an angle 1 from the normal incident position is defined by the following equation: OP=OP 1−OP =0=m 0 where m is the number of fringe shifts observed from the Michelson interferometer and 0 is the wavelength of the light in free space. OP=n2 d − d n2 2 n d 2 1 1n1 L−n1 d 1 1 −n 2 d 0−n 1 Ln 1 d 0=m 0 2 n2 2
2
=n 2 d −
d n1 2 n d 1n1 L−n1 d 1 2−n2 d 0−n1 Ln1 d 0=m0 2 n2 2 1
2
=
n1 d 2 d n1 2 − =m0 2 1 2 n2 1
Solving for m and setting n 1=1 for air: d 2 1 m= 1 1− 2 0 n2
Appendix B
Introduction to Snell's Law of Refraction and the Concept of Optical Path Length
SNELL'S LAW: Direction of propagation of a light wave can be thought of as a ray. When a ray of light propagating in a medium with refractive index n 1 is incident on another medium that has a refractive index n 2n1 it experiences a decrease in its speed of propagation and a shift its trajectory. This phenomena is described by Snell's law, given by Eq. (B.1), and Figure 1 shows how the ray trajectory is shifted from medium to medium. sin 1 v 1 n 2 = = sin 2 v 2 n 1 (B.1)
Where v 1 is the speed of light in medium 1 and v 2 is the speed of light in medium 2 and v 1v 2 .
n1
1
n2
2
Figure B.1. Shift in the direction of ray propagation due to difference in the refractive indices.
OPTICAL PATH LENGTH: Since the propagation velocity decreases in media with higher refractive indices, the same physical length (measured in m) is seen differently by a traveling light wave in mediums with different refractive indices. Therefore, a new expression that depends on the refractive index of the medium is needed to express the traveled distance. In Ray Optics, Optical Path Length (OPL) is used to express the distance traveled by a ray of light and it is given by Eq. (B.2)
OPL=n L
(B.2)
where n is the index of refraction and L is the physical distance in m. For example, OPL of the ray
shown in Figure B.2 that travels through two media with different refractive indices has an optical path of OPL=n1 L1n2 l n1 L2 .
Physical Length: Optical Length:
L1 n 1 L1
l n2 l
L2 n1 L2
n1
n2
n1
Figure B.2. OPL of a ray passing through two different media.
Appendix C
MATLAB Scripts Used to Perform Curve Fitting Routine to Find Refractive Indices
% ============================================= % Curve Fitting Routine to find refractive index % of the red glass % ============================================= %-------------- Measured Data ----------alpha_meas =(pi/180).*[0 2 4 6 8 10 12 14 16 18]; m_meas = [0 4 9 14 27 47 68 97 128 166]; %---------------------------------------n2 = 1:.001:1.9; lambda = 633e-9; %He-Ne Laser Wavelength. d = 0.00312; %Thickness of the Red Glass. m_array ={}; F =[]; for i = 1:length(n2) m =[]; for k = 1:length(alpha_meas) m = [m (1-1/n2(i))/lambda*d*alpha_meas(k)^2]; end m_array{i} = m; F =[F sum((m_meas - m).^2)/length(m_meas)]; %Claculating the error variance end plot(n2, F) title(['Plot of Error Variance. Min Error Found @ n2 = ',num2str(n2(find(F == min(F)))), 'for the Red Glass' ]) xlabel ('Index of Refraction'); ylabel('Error Variance') figure plot(180/pi.*alpha_meas,m_meas,'o') title('Red Glass - Counted Number of Fringes vs. Anlge') xlabel('Angle -Degrees') ylabel('Number of Fringes') hold on plot(180/pi.*alpha_meas,m_array{find(F == min(F))}) legend('Measured Data', 'Best Fit Curve') text(2,170,'n2 = 1.49')
%============================================== % Curve Fitting Routine to find refractive index % of the microscope slide. % ============================================= %-------------- Measured Data ----------alpha_meas =(pi/180).*[0 4 8 12 16 20 ]; m_meas = [0 2 7 17 33 50 ]; %---------------------------------------n2 = 1:.001:1.9; lambda = 633e-9; %He-Ne Laser Wavelength. d = 0.00134; %Thickness of the microscope slide (clear glass). m_array ={}; F =[]; for i = 1:length(n2) m =[]; for k = 1:length(alpha_meas) m = [m (1-1/n2(i))/lambda*d/2*alpha_meas(k)^2]; end m_array{i} = m; F =[F sum((m_meas - m).^2)/length(m_meas)]; %Claculating the error variance end plot(n2, F) title(['Plot of Error Variance. Min Error Found @ n2 = ',num2str(n2(find(F == min(F)))), ' for the Microscope Slide' ]) xlabel ('Index of Refraction'); ylabel('Error Variance') figure plot(180/pi.*alpha_meas,m_meas,'o') title('Microscope Slide - Counted Number of Fringes vs. Anlge') xlabel('Angle -Degrees') ylabel('Number of Fringes') hold on plot(180/pi.*alpha_meas,m_array{find(F == min(F))}) legend('Measured Data', 'Best Fit Curve') text(2,55,'n2 = 1.63')
REFRACTIVE INDEX MEASUREMENT USING THE MICHELSON INTERFEROMETER
Sahand Noorizadeh Charlie Hunter ECE 4501 – Georgia Institute of Technology NOVEMBER 2009
INTRODUCTION:
Refractive index is a measure of how much the speed of light (or other forms of waves) is reduced inside a medium compared to the speed of light in a vacuum. It is denoted by n and defined by Eq. (1)
n= c V
(1)
where c is the speed of light in vacuum and V is the speed of light in the medium. For example, the refractive index of light in air is about 1.0003 which means that the speed of light in air is very close to c . When light hits a boundary of two different indices of refraction, a portion of the incident wave is reflected while the rest is propagated through. The angle of propagation of the transmitted wave changes (bends) due to the change in refractive index. This behavior of light in materials with different refractive indices that makes waveguiding possible in optical fibers. Light propagates more slowly through media with higher refractive indices. Because of this, the optical path of light is equal to the physical distance times the refractive index of the medium, and therefore an optical distance is not the same as the physical distance. This property can be used in conjunction with a Michelson interferometer to measure the refractive index of a material with a known thickness. The Michelson interferometer was invented in early 1880's for the famous “failed” Michelson-Morely experiment to detect aether flow. In a modern Michelson interferometer, a coherent light beam, such as a laser, is split into two different beams and then reflected back to construct a single beam again. The newly joined beam forms an interference pattern with minima and maxima (fringes) that are exactly one wavelength apart. Introducing a material with a higher refractive index than air to the path of one of the split beams causes a change in the optical path. Observing the movement of these fringes as the optical path changes can give insight to the refractive index of that material. This lab report is concerned with setting up and using a Michelson interferometer to measure refractive indices of two glasses with known thickness. 1
THEORY:
In constructing a model to obtain the refractive index of a material with known thickness, two basic principles of optics were used and are briefly described in Appendix B.
THE MICHELSON INTERFEROMETER:
In 1887, before the discovery of light as a form of wave, Albert Michelson and Edward Morley conducted an experiment using a set-up similar to the one shown in Figure 1 to search for motion through aether. After not obtaining any result close to their expectation, the experiment was declared a failure. It was concluded later that their experiment had shown that the hypothesis of a stationary, luminiferous ether was not correct.
As shown in Figure 1, the Michelson interferometer uses a beam splitter to split a coherent light into two different beams and sends them in different directions. A pair of perpendicular mirrors
M irrors
Translation Stage
LASER Source M irrors
Beam Splitter
Screen
Optical Fringes
Figure 1. The Michelson interferometer. 2
mounted on translation stages are used to return the beams to the beam splitter at the same point to reconstruct the initial single beam. If the returning beams perfectly line up, an interference pattern will be formed in the reconstructed beam with minima and maxima as indicated by optical fringes observed on a screen. The gap between each fringe is exactly one wavelength (of the source beam). Moving any mirror pair on any of the interferometer's arms along their incident beams would cause the fringes to move and if they are displaced by a distance equal to the distance of two neighboring fringes, it would mean that the mirror pair was displaced by one wavelength.
APPARATUS
The objective of this lab was to measure refractive indices of a red glass and a microscope slide using the Michelson interferometer with a 633 nm He-Ne laser source. The apparatus shown in Figure 3 was assembled to achieve this objective. The first step of this experiment was to set up the Michelson interferometer of Figure 1 and adjust the position of the two mirror pairs to line up the returning split beams and observe the optical fringes on a screen. The next step was introducing a glass mounted on a rotary translation stage to the beam path of one of the interferometer's arms.
The idea was to use the fact that the overall OPL of the light would increase when it passes through the glass (that has a higher refractive index than air) and then rotate the glass to increase the distance that the light had to travel through the glass which would in turn increase the OPL and move the fringes. The increase in the traveling distance of a light beam through a material by rotating the material is shown in Figure 2. The rotary translation stage used in this experiment had a resolution of 1°.
Since counting the number of fringe shifts by eyesight was very difficult, the reconstructed beam was magnified and projected onto a photo multiplier with a pinhole on its aperture to detect the 3
minima and the maxima as the fringes moved. An adjustable speed chopper was placed on the path of the beam to periodically intersect the beam in order to make “the absence of the beam” as the oscilloscope reference point. The set-up of this experiment is shown in Figure 3.
l0
l1
l2
l3
1
2
3
Figure 2. Increase in the traveling distance of a light beam through a material by rotating the material. Mirrors
Glass Under Test 630nm He-Ne LASER
Rotary Translation Stage
Mirrors Beam Splitter Magnifier Oscilloscope
Adjustable Speed Chopper
Photo Multiplier Figure 3. The apparatus used to measure the refractive index of a glass with known thickness using the Michelson interferometer. 4
The job of the photomultiplier was to detect the incoming photons and generate an electric voltage proportional to the number of photons. This voltage was monitored with an oscilloscope. The minama (dark spots) of the interference pattern generated no voltage (reference level) and the maxima (bright spots) generated the highest negative voltage. A change of voltage from maximum negative to the reference level back to the maximum negative indicated one fringe shift. The speed (frequency) of the chopper had to be adjusted to obtain a periodic signal with minimum variation on its peaks.
DATA:
First, the red glass was mounted on the rotary stage and was set to the normal incident angle which was found by matching the beam reflection spots from the back of the red glass to the visible beam spots on the corresponding mirror pair. The angle of the rotary translation stage at this position was used as the reference angle (0°). The data collected for the red glass are listed in Table 1. The red glass intersected both the incoming beam from the beam splitter and the returning beam from the mirror pair. Table 1 – Rotation Angles of the Red Glass and the Corresponding Number of Fringe Shifts
Angle (Degrees) # of Fringe Shifts
0 0
2 4
4 9
6 14
8 27
10 47
12 68
14 97
16 128
18 166
The same process was repeated for a clear microscope slide but for this part of the experiment, only the returning beam from the mirror pair was intersected by the microscope slide. The data collected for the microscope slide are listed in Table 2.
Table 2 – Rotation Angles of the Microscope Slide and the Corresponding Number of Fringe Shifts
Angle (Degrees) # of Fringe Shifts
0 0
4 2
8 7
12 17
16 33
20 50
24 73
28 97
32 126
36 163
5
The thicknesses of the two glasses were measured with a micrometer and are listed in Table 3. Table 3 – Thickness Measurements
Material Red Glass Microscope Slide Thickness – mm 3.12 1.34
CALCULATIONS:
The last step in finding the refractive index of the two materials tested in this experiment was developing a theoretical model that could relate the number of fringe shifts to the rotation angle, thickness of the material, and of course the index of refraction. A model was developed based on the concept of Optical Path Length and its derivation is presented in detail in Appendix A. Eq. (2) and Eq (3) were developed for the red glass and the microscope slide respectively to model the number of fringe shifts as a function refractive index of the material n 2 , laser source wavelength 0 , rotation angle , and the thickness of the material d . m= d 2 1 1 1− 0 n2
2
(2)
d 1 m= 1 1− 2 0 n2
(3)
Eq. (2) is missing a factor of 1/2 because the red glass intersected both incoming beam from the beam splitter and the returning beam from the mirror pair which doubled the OPL of interest for the calculations.
To find the refractive index, a MATLAB® script was written to do the curve fitting routine by performing the following steps: 1. Generating an array of n 2 ranging from 1.000 to 1.900 with increments of 0.001. 6
2. Calculating m given by Eq. (2) and Eq. (3) for each value of n 2 for all angles of rotation. 3. Iterating through all sets of m generated in step 2 and comparing each one of them with the measured set of m and calculating the statistical error variance given by Eq. (4).
1 error= F n2 = N
i= N i=0
∑ mn2 meas.−mn 2 generated 2
(4)
4. Finding the value of n 2 that resulted in the least error variance.
Figure 4 and Figure 5 show the best fit curve to the measured data points of the red glass and microscope slide respectively. For the microscope slide, only data from rotation angles of less than 20 degrees were used to find the refractive index because in deriving the model, small angle approximation were used in applying the Snell's law.
R 1 8 0 n 2 1 6 0 1 4 0 1 2 0 N u m b e r o f F rin g e s 1 0 0 8 0 6 0 4 0 2 0 0 M B 2 4 6 A 8 1 0 n g le - D e a s u r e d e s t F it C 1 6 1 8 a t a u r v e D = 1 . 4 9 e d G la s s C o u n t e d N u m b e r o f F r in g e s v s .
A
0
1 2 1 4 e g r e e s
Figure 6. Best fit curve to the data obtained for the red glass. 7
M 6 0 n 2 5 0
ic r o s c o p e = 1 . 6 3
S
li d e
-
C
o u n t e d
N
u m
b e r
o f
F
r i n g
4 0 N u m b e r o f F rin g e s
3 0
2 0
1 0 M B 0 0 2 4 6 8 A 1 0 n g le e a s u r e d e s t F it C 1 8 2 0 a t a u r v e D
1 2 1 4 1 6 - D e g r e e s
Figure 7. Best fit curve to the data obtained for the microscope slide.
Table 4 lists the refractive indices of these two materials. The MATLAB® script used to generate these results is available in Appendix C.
Table 4 – Indices of Refraction Found by the Best Fit Curve Method
Material Red Glass Microscope Slide Index of Refraction 1.49 1.63
8
DISCUSSION:
Expected Results: Two different indices were obtained for each glass sample because two different
models were used: one that took into account the change in trajectory due to refraction and one that assumed a straight path through the material under test. When using the refraction model, the physical distance through the glass sample was less than the straight path model. The optical path length did not change between models ( the same counted fringes data was used) and optical path length is equal to the refractive index times the physical distance. Therefore, the refractive index was expected to be lower in the straight path model compared to the refraction model because the physical distance was smaller. For the microscope slide, it was expected that the refractive index would be close to 1.44 for the refraction model, which is the value for standard glass, and less than 1.44 for the straight path model. For the red glass, it was expected that the refractive index would be greater than 1.44 because of the dopants used in the glass to create the filter. So the refraction model was expected to yield a result very close to the actual refractive index, and the straight path model to yield a result less than the refractive index. Experimental Results: The refractive indices for the microscope slide were 1.37 for the straight
path model and 1.62 for the refraction model. The result from the refraction model was much larger than expected. The refractive index was less for the straight path model, and there was a significant difference between the two models. It is likely that the microscope glass has properties different than standard clear glass, perhaps added dopants, that increase the refractive index. The red glass refractive indices were 1.33 for the straight path model and 1.49 for the refractive model, very close to standard glass. Both of these values were less than expected and neither were greater than the clear glass microscope slide values. Again, the difference in the models was significant, with the refraction model resulting in the larger value. Dopants were used in the red glass 9
to create the filter effect, which would increase the refractive index. However, that filtering effect is dependent on wavelength. Because a red laser was used, it is likely that the wavelength of the source was in the pass band of the filter, causing the filter glass to act like standard clear glass: having a refractive index near 1.44. If an error occurred, it could have been one of two things: an error in the measurement, or an error in the model. The model worked according to expectation because the values of the refraction model were larger than the values of the straight path model, and all of the values were between 1.3 and 1.7. There were several places where an error could have occurred in the measurement process. Because the small angle approximation was used, all the data for angles greater than 20 degrees was ignored. For the microscope slide, that left the data to just 5 points, which may not have been enough to result in an accurate refractive index. Also, it is very difficult to create an exact 90 degree angle with the mirrors in the arms of the interferometer. When the refraction of light through the glass moved the output beam, it would return with a slightly different optical path if the mirrors were not precisely 90 degrees apart. This could have caused an extra fringe to be counted if the path was off by just one wavelength (633 nanometers). It is also possible that the measured thicknesses of the glass samples were not uniform throughout the sample, causing the thickness measurement to be inaccurate. Again, one wavelength difference would result in one fringe difference.
CONCLUSION:
An interferometer was used to measure the refractive index of two sample glass materials: a clear microscope slide and a red filter glass. This was done by measuring the fringe shifts of the beam output when the sample glass was rotated in the path of one of the interferometer arms. The shift is change in optical path, which is dependent on the refractive index of the material. Curve fitting models were developed and used to find the refractive index based on the data taken. For the microscope slide, 10
the results were higher than expected. For the red filter glass, the result were lower than expected. However, exploring the possibilities of what caused these results gave insight as to what could have happened. The microscope slide may have been made of a different material than originally assumed. The red glass filter most likely had a low absorption band at the wavelength of the source.
11
Appendix A
Derivation of the Theoretical Model for Calculating the Number of Fringe Shifts Based on Change in the Optical Path
d
n1 n2 2 1 2 n1 1 l3
l1
l2
d1
L
Figure A.1. Refraction of light ray inside a rotated glass.
From Figure A.1, the overall physical length of the light ray: L=l 1 d 1l 3 → l 3=L−l 1−d 1 l 2= d cos 1 and d 1=l 2 cos 1−2= d cos 1− 2 cos 2
and the optical path of the ray is:
OP=n1 l 1 n 2 l 2n1 l 3
=n 1 l 1n2
d n L−l 1−d 1 cos 1 1
=n 2
d [ cos 1 cos 2 sin 1 sin 2 ] d n 1 L−n 1 cos 1 cos 2 d n L−l 1−d 1 cos 1 1
=n 1 l 1n2
=n 2 d
tan2 n L−n1 d cos 2 −n 1 d sin 1 tan 2 sin 2 1
→ OP=n1 d tan 2 Using:
[
n2
2 1
1 −sin 1 n1 L−n1 d cos 1 sin 1 n
2
]
n 1 sin 1 =n 2 sin 2
Snell's law of refraction. n1 sin1 n2
tan 2=
sin 2 sin2 = = cos 2 1−cos 2
1−
n1
2
Trigonometry property of tan .
n2 2
sin 1
2
OP Can be written as: n1 sin 1 2 n2 n1 1 OP= n1 L−n1 d cos 1 2 2 sin 1 −sin 1 n2 n 1− 1 sin 2 1 n2 2 n1 d
[
]
The change in the optical path when the glass is rotated by an angle 1 from the normal incident position is defined by the following equation: OP=OP 1−OP =0=m 0 where m is the number of fringe shifts observed from the Michelson interferometer and 0 is the wavelength of the light in free space. OP=n2 d − d n2 2 n d 2 1 1n1 L−n1 d 1 1 −n 2 d 0−n 1 Ln 1 d 0=m 0 2 n2 2
2
=n 2 d −
d n1 2 n d 1n1 L−n1 d 1 2−n2 d 0−n1 Ln1 d 0=m0 2 n2 2 1
2
=
n1 d 2 d n1 2 − =m0 2 1 2 n2 1
Solving for m and setting n 1=1 for air: d 2 1 m= 1 1− 2 0 n2
Appendix B
Introduction to Snell's Law of Refraction and the Concept of Optical Path Length
SNELL'S LAW: Direction of propagation of a light wave can be thought of as a ray. When a ray of light propagating in a medium with refractive index n 1 is incident on another medium that has a refractive index n 2n1 it experiences a decrease in its speed of propagation and a shift its trajectory. This phenomena is described by Snell's law, given by Eq. (B.1), and Figure 1 shows how the ray trajectory is shifted from medium to medium. sin 1 v 1 n 2 = = sin 2 v 2 n 1 (B.1)
Where v 1 is the speed of light in medium 1 and v 2 is the speed of light in medium 2 and v 1v 2 .
n1
1
n2
2
Figure B.1. Shift in the direction of ray propagation due to difference in the refractive indices.
OPTICAL PATH LENGTH: Since the propagation velocity decreases in media with higher refractive indices, the same physical length (measured in m) is seen differently by a traveling light wave in mediums with different refractive indices. Therefore, a new expression that depends on the refractive index of the medium is needed to express the traveled distance. In Ray Optics, Optical Path Length (OPL) is used to express the distance traveled by a ray of light and it is given by Eq. (B.2)
OPL=n L
(B.2)
where n is the index of refraction and L is the physical distance in m. For example, OPL of the ray
shown in Figure B.2 that travels through two media with different refractive indices has an optical path of OPL=n1 L1n2 l n1 L2 .
Physical Length: Optical Length:
L1 n 1 L1
l n2 l
L2 n1 L2
n1
n2
n1
Figure B.2. OPL of a ray passing through two different media.
Appendix C
MATLAB Scripts Used to Perform Curve Fitting Routine to Find Refractive Indices
% ============================================= % Curve Fitting Routine to find refractive index % of the red glass % ============================================= %-------------- Measured Data ----------alpha_meas =(pi/180).*[0 2 4 6 8 10 12 14 16 18]; m_meas = [0 4 9 14 27 47 68 97 128 166]; %---------------------------------------n2 = 1:.001:1.9; lambda = 633e-9; %He-Ne Laser Wavelength. d = 0.00312; %Thickness of the Red Glass. m_array ={}; F =[]; for i = 1:length(n2) m =[]; for k = 1:length(alpha_meas) m = [m (1-1/n2(i))/lambda*d*alpha_meas(k)^2]; end m_array{i} = m; F =[F sum((m_meas - m).^2)/length(m_meas)]; %Claculating the error variance end plot(n2, F) title(['Plot of Error Variance. Min Error Found @ n2 = ',num2str(n2(find(F == min(F)))), 'for the Red Glass' ]) xlabel ('Index of Refraction'); ylabel('Error Variance') figure plot(180/pi.*alpha_meas,m_meas,'o') title('Red Glass - Counted Number of Fringes vs. Anlge') xlabel('Angle -Degrees') ylabel('Number of Fringes') hold on plot(180/pi.*alpha_meas,m_array{find(F == min(F))}) legend('Measured Data', 'Best Fit Curve') text(2,170,'n2 = 1.49')
%============================================== % Curve Fitting Routine to find refractive index % of the microscope slide. % ============================================= %-------------- Measured Data ----------alpha_meas =(pi/180).*[0 4 8 12 16 20 ]; m_meas = [0 2 7 17 33 50 ]; %---------------------------------------n2 = 1:.001:1.9; lambda = 633e-9; %He-Ne Laser Wavelength. d = 0.00134; %Thickness of the microscope slide (clear glass). m_array ={}; F =[]; for i = 1:length(n2) m =[]; for k = 1:length(alpha_meas) m = [m (1-1/n2(i))/lambda*d/2*alpha_meas(k)^2]; end m_array{i} = m; F =[F sum((m_meas - m).^2)/length(m_meas)]; %Claculating the error variance end plot(n2, F) title(['Plot of Error Variance. Min Error Found @ n2 = ',num2str(n2(find(F == min(F)))), ' for the Microscope Slide' ]) xlabel ('Index of Refraction'); ylabel('Error Variance') figure plot(180/pi.*alpha_meas,m_meas,'o') title('Microscope Slide - Counted Number of Fringes vs. Anlge') xlabel('Angle -Degrees') ylabel('Number of Fringes') hold on plot(180/pi.*alpha_meas,m_array{find(F == min(F))}) legend('Measured Data', 'Best Fit Curve') text(2,55,'n2 = 1.63')
