Particle Size Analysis by Laser Diffraction Method.
IAS 2013- Abhishek
1. Indian Institute of Space Science
and Technology
Department of Physics
Project Report
Author:
Surya Abhishek Singaraju
IMSc 2nd year,
University of Hyderabad
Supervisor:
Dr.C.S.Narayanamurthy
Professor and Head,
Department of Physics, IIST
2. Investigation on Some Optical
Interferometers using Coherence and
Geometric Phase
July 15, 2013
1
4. 1 Declaration
I,Surya Abhishek Singaraju, a student of the University of Hyderabad study-
ing Integrated MSc 2nd year(11ICMC13), and an Indian Academy of Sciences
2013 summer fellow with application number PHYS1530 hereby declare that
all the work done under the topic ”Investigation on Some optical Interfer-
ometers using Coherence and Geometric Phase” is done by me under the
guidance of Prof.C S Narayanamurthy, Professor and Head,Department of
Physics,Indian Institute of Space Science and Technology in his Adaptive
Optics labaratory from 15 May 2013 to 15 July 2013.
Place:
Date:
Signature(Supervisor): Signature(Summer Fellow):
3
5. 2 Acknowledgements
Firstly,I thank my parents who always supported me regardless of my achieve-
ments without which I would not have been into science.I thank my sis-
ter,Gayathri for giving nice valuable suggestions from her research experi-
ences.
I thank my cousin,Sudhir and my friends, Vamsi, Ritu, Divya, Harish, Sahithi,
Vivek, Satwik, Nithin, Dinesh, Praneeth and all other school and college
mates who stayed in touch with me,making sure I didnot miss home for the
two months.
I also thank Mrs Jayasree for welcoming me to their home and helping me
whenever I needed.
I thank my lab mates,Remya,Nikhil,Richa,Vinu and Anand for the wonderful
company in the laboratory.
I also thank the IIST canteen and maintenance staff for providing me with
a comfortable stay for two months.
Lastly, I thank my guide,Prof.C.S.Narayanmurthy who always encouraged
me to try new things and helped me through constant discussions.
4
6. 3 Abstract
Optical Coherence and phase are integral part of interferometry and play a
crucial role in the same. In this project, we investigated the effect of op-
tical coherence on interferometry for different light sources in Fabry Perot
interferometer and the effect of Geometric(Pancharatnam) phase on interfer-
ometry in various amplitude splitting interferometers like Michelson,Sagnac
and MachZehnder interferometers. Also,in these experiments,Pancharatnam
phase is observed even in the presence of only a single phase retarder. De-
tailed experimental analysis is carried out.
4 Multiple Beam Interferometry
4.1 Introduction
Optical Interference corresponds to the interaction of two or more light waves
yielding a resultant irradiance that deviates from the sum of the component
irradiance.[1] The most important conditions for optical interference are that
the light waves should be of the same wavelength and should be coherent.
Optical Coherence is the attribute of two or more waves, or parts of a wave,
whose relative phase is nearly constant during the resolving time of the ob-
server.[2]. It is related to the stability or predictability of the phase. Coher-
ence is classified into temporal and spacial Coherence. Temporal Coherence
describes the relation between the waves in terms of time difference whereas
Spacial coherence describes the same as a function of distance. The influence
of coherence is studied in Fabry Perot etalon, a multiple beam interferometer
using different optical sources.
4.2 Fabry Perot Interferometer
Fabry Perot Interferometer consists of two parallel mirrors which are used
to produce multiple beam interference. When the mirrors are held fixed and
adjusted to be parallel, it is called an etalon.It also consists of a convex lens
which is used to collimate the beam on to the mirrors and another focusing
lens.The etalon setup is as seen in Fig.1.
5
7. Fig.1. Schematic of the Fabry perot etalon
In this Fabry Perot setup, the influence of optical coherence is studied
using three light sources: Helium-Neon(He-Ne) laser, Sodium light and Mer-
cury light. The Etalon spacing,d(Fig.1) is varied from 1mm to 17mm and
at each value of the spacing, the three sources are used to check accuracy
at the corresponding etalon spacing. A Charge Coupled Device(CCD) cam-
era is used to view the interference pattern and IMAGE J software is used
to analyse the pictures. It is observed that for large etalon spacing(12mm
to 17mm), He-Ne laser is precise and the calculated value of the spacing
is best matching with the real value for He-Ne laser at the value.This is
because He-Ne laser has high coherence length. For small spacing(1mm to
4mm),Mercury(White) light gave accurate values of spacing from the calcu-
lations which were matching the real values better than for other sources.
Fringes are not observed after a spacing of 9mm.Sodium light is most accu-
rate for intermediate values(4mm to 11mm). Fringes are not observed for
Sodium light after a spacing of 14mm. Pictures of the fringes observed for
the different light sources were taken while varying the etalon spacing.
Fig.2. Fringes observed for He-Ne laser source
6
8. Fig.3. Fringes observed for Sodium source
Fig.4. Fringes observed for Mercury source
Figures 2 to 4 are the fringes observed at an etalon spacing of 5mm for
He-Ne laser,Sodium vapour and Mercury vapour lights respectively. It is
clear from the three pictures that Sodium source has very clear and more
number of fringes at 5mm spacing. He-Ne laser has lesser number of fringes
which increased as the etalon spacing increased. But for mercury light, the
fringes became more and more distorted as the spacing increased.
4.2.1 Results and Discussions
The values of the radii of the fringes are observed at every etalon spacing for
each light source. Starting from 1mm, the etalon spacing was increased till
17mm and the number of fringes and their radii are measured correspondingly
and can be seen in the tables below. Tables 1,2 and 3 give measurements of
radius of fringes with respect to etalon spacing for He-Ne laser, Sodium light
and Mercury source respectively.
7
10. From tables,1,2 and 3 it can be seen that as the etalon spacing increased,
the number of fringes for He-Ne laser increased but for sodium and mercury
lights, fringes could not be seen after 14mm and 9mm respectively. This
sensitivity of fringes is plotted in Plot.1 with respect to etalon spacing for
laser, Sodium lamp and Mercury source.
Plot 1: Comparisons in Fabry Perot Setup
In the plot, the green curve represents mercury light while the red and
blue curves represent sodium and He-Ne laser sources respectively. From
the graph, we can see that upto an etalon spacing of around 2.5mm, white
light has highest number of fringes(10). From 5 to 7mm, sodium light has
most fringes(16) and from thereon, He-Ne laser has the maximum fringes. At
15mm etalon spacing, the number of fringes for He-Ne laser is maximum(35)
followed by Sodium(25).
4.3 Result
From the above graph and calculation of spacing from the fringe width and
wavelength, it is observed that for a spacing of 1-3mm,Mercury(white) light
9
11. gave better readings and so did Sodium light for spacing of 4-8mm and He-Ne
laser is best suited for spacing beyond 8mm.So for small spacings white light
is best suited since its coherence is small and for large spacing, He-Ne laser
is best since its coherence is large.For medium spacing, Sodium lamp suits
the best.
In interferometry, phase difference can be obtained through different means.It
can be obtained by having a path difference between the interfering waves
as seen in the above Fabry Perot interferometer. But for the waves without
a path difference, the phase difference can be obtained by introducing cer-
tain objects in the path of the wave. Geometric Phase interferometry is an
example of such a method.
5 Geometric Phase Interferometry
5.1 Introduction
Geometric phase interferometry is a method of introducing a phase differ-
ence between the waves without the need of path difference and by just
creating few changes in the polarisation states of the light waves. When a
light is taken from one polarisation state through few changes and brought
back along the same path to it’s original polarisation state, it is still not in
the same state as it was initially. It undergoes a phase shift. This phase
shift is known as Panchartnam phase[3]. This phenomenon is similar to
a cyclic adiabatic process where the system comes to the initial state but
with a energy change. This phase shift can be measured from interference
experiments.Many experimental and theoritical attempts have been done suc-
cessfully to prove the pancharatnam phase for polarised light and to plot it
on the Poincare sphere.[4]-[14]. While the Pancharatnam phase was mostly
proved for cyclic polarisation state changes, it was also proved to be true
even for non cyclic changes.[15]. We report experimental investigation of
the Pancharatnam phase for different experimental setups with certain in-
teresting changes in them. We did the experiments in amplitude-splitting
inteferometers like Michelson,Sagnac and Mach Zender Interferometers by
using quarter wave plates and half wave plates. We tried to solve the mathe-
matics of the experiments using Jones matrices[16]-[17] as it was done earlier
and to plot it on the poincare sphere. The Jones matrices of each object
namely the polariser, Quarter Wave plates(QWP), Half wave plates(HWP)
and the mirror were taken into account[18]-[21] and the resultant matrices
for the setups were calculated.
The jones matrices of few objects are listed below
10
12. Jones matrix for a mirror
J =
1 0
0 −1
(1)
Jones matrix for Linear polarizer with axis of transmission vertical
J =
0 0
0 1
(2)
Jones matrix for a rotatable waveplate which can be rotated at an angle θ to the initial polarisation axis and creates
a phase difference of φ.[18]
J =
cosφ/2 + isinφ/2cos2θ isinφ/2sin2θ
isinφ/2sin2θ cosφ/2 − isinφ/2cos2θ
(3)
For a quarter wave plate which induces a phase difference of π/2, the Jones matrix
J =
1
√
2
1 + icos2θ isin2θ
isin2θ 1 − icos2θ
(4)
For a halfwave plate which induces a phase difference of π on the light wave, the Jones matrix
J =
cos2θ sin2θ
sin2θ −cos2θ
(5)
This matrix information is used in the determination of the output ma-
trices in each of the following interferometers.Experiments are carried out
using Michelson, Sagnac and Mach Zehnder interferometers.
5.2 Michelson Interferometer
The basic setup is simple and consists of two mirrors and a beam splitter.The
figure is seen below(Fig.5)
11
13. Fig.5. Basic michelson setup
The light from the source(He-Ne laser) is split into two beams by the
beam splitter and these are again reflected at the respective mirrors.Since
He-Ne laser is a coherent source,both these reflected beams interfere and this
is seen at the detector.The Pancharatnam phase can be created in one of the
beams and to achieve this, a polariser and quarter wave plates, one at a fixed
angle of 45 degrees to the initial polarisation and the other rotatable were
used[22].A photodetector is used to detect the intensity variation.(Fig.6)
However, it is observed that even when QWP 1 is not present also there is
Pancharatnam phase.(Fig.7) and when both QWPs are on either arms(Fig.8)
Fig.6.Schematic of expt with 2 QWPs Fig.7.Expt with one QWP
12
14. Fig.8.Schematic of another expt done with two QWPs on two arms
The Schematic of the closed path followed by the polarisation state of the
light for Fig.6 is given on the poincare sphere below.The initial light reach-
ing the Quarter wave plate 1 is linearly polarised and is represented by point
A on the Poincare sphere.This light once passing through the QWP1 gets
circularly polarised and is shown as point B on the poincare sphere.After
passing through QWP2(rotatable),it again becomes linearly polarised(point
C on poincare sphere).The light is then reflected from mirror M1 and retraces
its path through the Quarter wave plates thus completing the loop through
D to A on the poincare sphere.The complete loop is ABCDA.
If θ and φ are the polar and azimuthal coordinate angles of any point on
the Poincare sphere,respectively, the 2 X 2 polarization or coherence matrix
J, normalized to unit trace, representing an arbitrarily (fully) polarized plane
wave[23] is of the form
Eqn.6. Coherence matrix of arbitrarily polarised plane wave
J =
1
2
1 + cosθ sinθeiθ
sinθe−iθ
1 − cosθ
(6)
13
15. Fig.9. Poincare sphere schematic of the Pancharatnam phase in Michelson setup
From the knowledge of the Jones matrices, the matrix of each experiment
is solved i.e for each arm in the Michelson setup.
The Jones matrix for the arm containing the single QWP in Fig.7(Appendix.1)
Eqn.7 Jones matrix for single QWP
J =
sin2
2θ + icos2
2θ −sin4θ
2
−sin4θ
2
−sin2
2θ − icos2
2θ
(7)
The Jones matrices for each arm in Fig.8 containing Quarter wave plates
is as following:
For the arm containing QWP fixed at 45 degrees to the initial polarisation
axis, the Jones matrix is written as
J =
1 0
0 −1
(8)
This is same as the matrix of a mirror.(see Appendix.1)
And for the arm containing the rotatable quarterwave plate, the matrix
is same as Eqn.7 and is equal to
J =
sin2
2θ + icos2
2θ −sin4θ
2
−sin4θ
2
−sin2
2θ − icos2
2θ
(9)
14
16. 5.2.1 Results and Discussions
In Eqn.8, we have seen that the matrix of the arm containing QWP fixed at
45 degrees to the original polarisation state is same as that of mirror. So,
this means the experimental setups in Fig.7 and Fig.8 are the same.To prove
the pancharatnam phase with a single quarterwave plate, intensity measure-
ments are taken for every 10 degree rotation of the rotatable quarterwave
plate for 360 degrees and are tabulated in Table.4 below.
15
18. The graph of the angle of rotation of the quarterwave plate in degrees
as it is rotated for 360 degrees and the resulting intensity of the interference
pattern in mA and mW is plotted for the three different setups namely with
both QWPs on same arm i.e the blue curve for which the intensity is plotted
in mA, with a single QWP on the red curve and with both the QWPs on
either arm seen as the green curve for which the intensity is plotted in mW.
Plot.2.Geometric phase in Michelson setup
From the graph, it is clear that even the experiment with a single QWP is
showing Pancharatnam phase. Also it is shown that the quarter wave plate
fixed at 45 degrees and a mirror together act as a simple mirror reflection only.
5.3 Sagnac Interferometer
In the michelson interferometer, there could be an error in the geometric
phase due to a possible change in the length of one of the optical paths lead-
ing to a change in phase(dynamic phase).This problem is avoided by using a
17
19. sagnac interferometer in which the two beams traverse the same optical path
in opposite senses. The phase difference between the beams thus depends
only on the Pancharatnam phase[24].Sagnac Interferometer is similar to the
Michelson interferometer but the mirrors are tilted in such a way that they
face each other so that both the beams reflected on either of them travel
along the same path.
Fig.10 Basic Sagnac setup
In the sagnac setup, two QWPs and a Half Wave Plate(HWP) in the
middle are kept along the beam between the two mirrors(Fig.11).In this
setup too, it is very clear that even in the absence of the two QWPs, there
is Pancharatnam phase(Fig.12).
Fig.11 Schematic of expt with QWPs,HWP Fig.12 Expt with only HWP
18
20. The geometric phase is set on the path between both the mirrors using
quarter wave plates and a half wave plate(Fig.11).For detecting intensity,a
photodetector is used.The operation of the interferometer is given on the
poincare sphere below.Initially,the p-polarised light which is linear is con-
verted to left circular polarisation state by QWP1 and is represented by
point A1 on the poincare sphere.Then, the half wave plate whose axis is at
an angle θ to the principal axis of QWP1 shifts this to right circular polari-
sation state.Then the QWP2 brings the polarisation state back to the initial
linear polarisation.Thus the pancharatnam loop on the poincare sphere for
p-polarised light is A1SA2NA1. Similarly for the s-polarised light it is
B1SB2NB1.This arrangement gives rise to an additional 2θ phase differ-
ence thus give=ing a total phase difference of 4θ.[24]-[25]
Fig.13.Poincare sphere schematic of the Pancharatnam phase in Sagnac setup
The Jones matrices of the experimental setups with only halfwave plate
are found out.(see Appendix.2)
The jones matrix for one arm(p arm) in sagnac setup with only HWP
J =
cos2θ −sin2θ
−sin2θ −cos2θ
(10)
19
21. For the other arm(s arm) in sagnac setup with only HWP, Jones matrix
J =
cos2θ sin2θ
sin2θ −cos2θ
(11)
Jones matrix for 2 QWPs in sagnac setup
J =
0 −i
−i 0
(12)
5.3.1 Results and Discussions
Even without the two quarter wave plates in the original setup, Pancharat-
nam phase is observed. This is just by rotating the halfwave plate without
the quarter wave plates. To prove the pancharatnam phase with only half
wave plate, intensity measurements are taken for every 10 degree rotation of
the rotatable quarterwave plate for 360 degrees and are tabulated in Table.5
and Table.6 below.
Table 5: Geometric Phase with 2 QWPs and 1 HWP
θ Intensity(µA) θ Intensity(µA) θ Intensity(µA) θ Intensity(µA)
0 4.7 100 3.9 190 1.9 280 5.4
10 2.2 110 7.2 200 0.9 290 8.9
20 1 120 10.7 210 0.9 300 11.8
30 0.8 130 13.4 220 1.2 310 14.3
40 1.2 140 14.4 230 1.3 320 14.8
50 1.5 150 13.6 240 1.1 330 13.2
60 1.4 160 11.2 250 0.9 340 10.1
70 1.1 170 7.4 260 1.3 350 6.9
80 1.1 180 4.1 270 3 360 3.4
90 2
20
22. Table 6: Geometric Phase with only HWP
θ Intensity(µA) θ Intensity(µA) θ Intensity(µA) θ Intensity(µA)
0 99.8 100 28.1 190 19 280 23.5
10 24.7 110 7.6 200 5.1 290 7.9
20 5 120 43.8 210 40.4 300 42.1
30 43.2 130 116.1 220 115.5 310 114.9
40 112.1 140 186.9 230 184 320 189.7
50 189.1 150 200 240 200 330 200
60 200 160 200 250 200 340 200
70 200 170 154.2 260 156.3 350 142.7
80 165.8 180 78.6 270 83.7 360 62.7
90 89.8
The graph of the angle of rotation of the halfwave plate in degrees as it is
rotated for 360 degrees and the resulting intensity of the interference pattern
in µA is plotted for the two different setups namely with HWP between two
QWPs on the blue curve and with only HWP as the red curve.
Plot.3.Geometric phase in Sagnac setup
21
23. From the graph, it is clear that even the experiment with only the HWP
is showing Pancharatnam phase. In fact it is showing a better reult as the
maximum and minimum intensities are observed clearly.
5.4 Mach Zehnder Interferometer
The next experiment done is to measure the Pancharatnam phase in a Mach
Zehnder interferometer. This was done previously by Piotr Kurynomski,Wladyslaw
and Szarycz[26]. In a Mach Zehnder interferometer which is used to measure
the path shift caused by a medium(Fig.14), the Pancharatnam phase interfer-
ometry was performed using quarter wave plates and half wave plate(HWP).
The setup is such that the path travelled by the two beams split by a beam
splitter is the same.
Fig.14 Basic Mach Zehnder setup
In the Mach Zehnder setup, two quarter wave plates with a half wave plate
in the middle are inserted(Fig.15). However even in this setup, it is clear that
there is Panchratnam phase even without the quarter wave plates(Fig.16).
Fig.15.Schematic of expt with QWPs,HWP Fig.16 Expt with only HWP
22
24. On one arm, Pancharatnam phase is setup by introducing quarter wave
plates and a half wave plate and the two beams are passed through another
beam splitter and then allowed to interfere. There is no path difference be-
tween the two light beams but there is a phase shift on one arm due to the
Quarter wave plates and the half wave plate(Fig.15). Each of these plates
transform the polarisation state into B,C and come back to A in the figure
shown below. Regardless of the intermediate states B and C the outgoing
light has same polarisation as input light A thus following the loop ABCA.
However, there is a phase shift and following Pancharatnam,we can calculate
phase shift as a half of the solid angle designated by the geodesic triangle
on the Poincare sphere[26]. The Pancharatnam triangle to calculate the ge-
ometric phase on the Poincare sphere for this is shown in Fig.17
Fig.17.Pancharatnam’s triangle on Poincare sphere in Mach Zehnder setup
The Jones matrices for the Mach Zehnder experiment done are found.(see
Appendix.3)
Jones matrix of arm with single halfwave plate
J =
cos2θ −sin2θ
sin2θ cos2θ
(13)
23
25. Jones matrix of the arm with only two QWPs(see Appendix.3)
J =
0 −i
i 0
(14)
5.4.1 Results and Discussions
Even without the two quarter wave plates in the original setup, Pancharat-
nam phase is observed. This is just by rotating the halfwave plate without the
quarter wave plates. To prove the pancharatnam phase with only half wave
plate, intensity measurements are taken for every 10 degree rotation of the
rotatable quarterwave plate for 360 degrees and are tabulated in Table.7(2
QWPs and HWP) and Table.8(only HWP) below.
Table 7: Geometric Phase with 2 QWPs and 1 HWP
θ Intensity(mW) θ Intensity(mW) θ Intensity(mW) θ Intensity(mW)
0 3.32 100 4.138 200 5.184 300 4.153
10 3.189 110 4.018 210 4.975 310 4.363
20 3.394 120 3.896 220 4.88 320 4.965
30 3.615 130 3.92 230 4.5 330 4.875
40 3.855 140 4.072 240 4.793 340 5.183
50 4.03 150 4.252 250 5.08 350 5.443
60 4.08 160 4.3 260 5.28 360 4.994
70 3.968 170 4.618 270 5.18
80 4.076 180 5.034 280 4.719
90 4.125 190 5.337 290 4.472
24
26. Table 8: Geometric Phase with only one HWP
θ Intensity(mW) θ Intensity(mW) θ Intensity(mW) θ Intensity(mW)
0 2.623 100 3.159 200 3.062 300 2.965
10 2.696 110 3.155 210 2.99 310 2.95
20 2.732 120 3.101 220 2.943 320 3.086
30 2.924 130 3.073 230 2.896 330 3.128
40 2.836 140 3.003 240 2.821 340 3.047
50 2.804 150 2.786 250 2.896 350 3.105
60 2.84 160 2.89 260 2.912 360 3.185
70 2.888 170 2.998 270 3.005
80 2.994 180 3.092 280 3.064
90 3.103 190 3.108 290 3.021
The graph of the angle of rotation of the halfwave plate in degrees as it is
rotated for 360 degrees and the resulting intensity of the interference pattern
in mW is plotted for the two different setups namely with HWP between two
QWPs on the blue curve and with only HWP as the red curve.
25
27. Plot.4.Geometric phase in MachZehnder setup
From the graph, it is clear that even the experiment with only the HWP
is showing Pancharatnam phase.Thus, in the three different interferometers,
it is seen that even if the quarter wave plates fixed at 45 degrees are not
present, there is Pancharatnam phase.
5.5 Results
In the geometric phase experiments done with Michelson,Sagnac and Mach
Zehnder interferometers, it is observed that geometric phase is observed even
when the Quarter wave plates at 45 degrees are not present.
In the experiments done in Michelson, Sagnac and Mach Zehnder interferom-
eters,the Quarter wave plates fixed at 45 degrees were responsible for circular
polarisation and the construction of a Pancharatnam triangle was possible.
However when there is only one phase retarder present, Pancharatnam phase
can still be constructed.It might look like the following
26
28. Fig.18.Schematic of the closed path followed by polarisation states of light on the Poincare sphere
Figure.15 is the construction of Pancharatnam loop on the poincare sphere
to calculate the geometric phase. The light which is initially linearly po-
larised(represented by A in the above Poincare sphere) before entering the
Quarter wave plate is elliptically polarised on coming out of the Quarter wave
plate and is represented by point B on the Poincare sphere. After reflection
at the mirror, the light retraces its path through C to A. Thus the closed
loop ABCA is constructed on the Poincare sphere thus concluding that this
is indeed a Pancharatnam Phase.
27
29. 6 Conclusion
In part one, i.e the multiple beam interferometry we concluded that as the
etalon spacing increased, the accuracy of Mercury light and Sodium light
decreased and He-Ne laser increased thus proving that white light is more
accurate only for small spacing, Sodium light for medium spacing and He-Ne
laser for longer spacings.In the second part, i.e in the Geometric Phase inter-
ferometry, we could prove that Pancharatnam phase could be observed even
in the presence of a single phase retarder.As seen in Figure.15, Pancharat-
nam phase is possible even with a single phase retarder. This might be due
to the change in polarisation introduced by mirror.[27]
28
30. 7 Appendix
7.1 Appendix.1
In the Michelson experiment,the Jones matrix for the arm containing the
single QWP as seen in Fig.7 can be found by multiplying the matrices of
each component i.e the rotatable quarterwave plate, mirror and again the
quarterwave plate.
J =
1
√
2
1 + icos2θ isin2θ
isin2θ 1 − icos2θ
×
1 0
0 −1
×
1
√
2
1 + icos2θ isin2θ
isin2θ 1 − icos2θ
So, for the arm containing only one quarterwave plate whose axis is ro-
tatable with respect to the initial axis of polarisation,the matrix reduces to
J =
sin2
2θ + icos2
2θ −sin4θ
2
−sin4θ
2
−sin2
2θ − icos2
2θ
(15)
If the axis of this quarterwave plate is fixed at 45 degrees, then the matrix
for the combination of only the quarterwave plate fixed at 45 degrees and
mirror is given by
J =
1
√
2
1 i
i 1
×
1 0
0 −1
×
1
√
2
1 i
i 1
=
1 0
0 −1
(16)
Thus, the matrix of a mirror and a quarterwave plate infront of a mirror is
the same.
7.2 Appendix.2
In the Sagnac experiment,the Jones matrix for the arm containing the single
HWP as seen in Fig.12 can be found by multiplying the matrices of each com-
ponent i.e the mirror, rotatable halfwave plate, and again the other mirror.
J =
1 0
0 −1
×
cos2θ sin2θ
sin2θ −cos2θ
×
1 0
0 −1
29
31. Hence, the matrix for the p-arm with only halfwave plate is written as,
J =
cos2θ −sin2θ
−sin2θ −cos2θ
(17)
For s-arm, there is a π difference. So, the matrix reduces to
J =
cos2θ sin2θ
sin2θ −cos2θ
(18)
For the s-arm with two quarterwave plates, the matrix is written as,
J =
1 0
0 −1
×
1
√
2
1 i
i 1
×
1
√
2
1 i
i 1
×
1 0
0 −1
=
0 −i
−i 0
(19)
7.3 Appendix.3
In the Mach Zehnder experiment,the Jones matrix for the arm containing
the single HWP as seen in Fig.16 can be found by multiplying the matrices
of each component i.e the rotatable halfwave plate and the mirror.
J =
cos2θ sin2θ
sin2θ −cos2θ
×
1 0
0 −1
=
cos2θ −sin2θ
sin2θ cos2θ
(20)
Similarly for the arm containing only two quarterwave plates, the matrix
formation is equal to
J =
1
√
2
1 i
i 1
×
1
√
2
1 i
i 1
×
1 0
0 −1
=
0 −i
i 0
(21)
30
32. 8 References
1.Optics,Fourth Edition by Hecht,Eugene
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