Mach–Zehnder
Interferometer
Eswar Gowtham 21MS165
Group - B10
ABSTRACT
The Mach-Zehnder interferometer is a versatile optical instrument widely used
to study interference phenomena. In this experiment, we investigate the formation of circular interference fringes observed on two distinct screens, which
exhibit a relative phase difference of π due to the splitting of an incident beam.
The interference pattern arises from the coherent recombination of these beams,
allowing for the precise analysis of phase shifts introduced in the optical path.
Furthermore, we explore the determination of the refractive index of air at
standard atmospheric pressure by introducing controlled variations in pressure
within a specialized pressure cell. By analyzing the resulting displacement of
interference fringes, we quantify the change in optical path length and extract
the refractive index using established theoretical relations. This study provides
insights into wavefront division interference and its application in precision measurements of refractive indices in gaseous media.
Contents
1 Introduction
1
2 Theory of Operation
2
3 Experimental Procedure
3.1 Setup Alignment and Calibration . . . . . . . . . . . . . . . . .
3.2 Observation of Circular Fringes . . . . . . . . . . . . . . . . . .
3.3 Measurement of the Refractive Index of Air . . . . . . . . . . .
2
2
3
3
4 Results
4
5 Error Analysis
6
6 Discussion
6.1 Explanation behind Linear and Circular Fringes . . . . . . . .
6.2 Phase Acquisition and Interference Patterns . . . . . . . . . . .
6.3 Sources of Error . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Error Mitigation Strategies . . . . . . . . . . . . . . . . . . . .
7
7
7
8
8
7 Conclusion
8
1
Introduction
Interferometry is a powerful technique in optics used to analyze wavefronts, measure small displacements, and determine the refractive index of materials with
high precision. The Mach-Zehnder interferometer is a well-known device that
demonstrates interference by splitting a coherent light beam into two distinct
paths, manipulating their relative phases, and recombining them to produce an
interference pattern. This experiment focuses on studying interference fringes
and utilizing the interferometer to measure the refractive index of air as a function of pressure.
In a Mach-Zehnder interferometer, a coherent light source, typically a laser, is
split into two beams by a partially reflective beamsplitter. Each beam traverses
a separate optical path before recombining at a second beamsplitter, leading to
interference. The phase difference between the two beams determines the nature of the interference observed at the two output ports. If both paths are equal
in optical length, constructive interference occurs at one detector and destructive interference at the other. However, any variation in optical path length due
to phase shifts, reflections, or changes in refractive index alters the interference
pattern.
A critical aspect of this experiment is understanding the effect of phase shifts
upon reflection. When light reflects from a medium with a higher refractive index, it experiences a phase shift of π, whereas transmission does not introduce
a phase shift. This corrects the apparent paradox of energy conservation in the
interferometer and ensures accurate analysis of interference effects.
The second objective of the experiment involves determining the refractive index of air at different pressures. Since air is a dispersive medium, its refractive
index varies with pressure following the Lorenz-Lorentz equation:
n2 − 1 X
=
Ri ρi ,
n2 + 2
i
where Ri is the specific refraction and ρi is the partial density of each gaseous
component in air. By inserting a sealed pressure cell in one arm of the interferometer, controlled pressure variations induce changes in the optical path length,
causing shifts in the interference fringes. The refractive index change ∆n is determined from the fringe shift ∆m using:
∆n ∆mλ0 /d
=
,
∆P
∆P
where λ0 is the laser wavelength in vacuum, d is the length of the pressure cell,
and ∆P is the pressure difference. By plotting n as a function of pressure and
extrapolating to standard atmospheric conditions, we obtain an accurate value
of air’s refractive index at 1 atm.
This experiment not only reinforces fundamental concepts in wave optics and
1
interferometry but also demonstrates the practical application of interference in
high-precision refractive index measurements. The results obtained have implications for atmospheric optics, precision metrology, and experimental physics.
2
Theory of Operation
The Mach-Zehnder interferometer is a device designed to demonstrate and analyze interference effects by splitting a coherent light beam into two paths. The
fundamental working principle involves a beamsplitter that divides the incident
beam into two coherent components traveling through different optical paths
before recombining at a second beamsplitter.
When light encounters a dielectric-coated beamsplitter, a portion of the beam is
transmitted, while the rest is reflected. Importantly, reflection from a higher refractive index medium introduces a phase shift of π, whereas transmission does
not affect the phase. The second beamsplitter ensures that the two beams interfere either constructively or destructively, depending on their relative phase
difference, given by:
2π(l1 − l2 )
,
δ=
λ
where l1 and l2 are the optical path lengths in the upper and lower arms of the
interferometer, respectively.
If the path difference l1 − l2 is an integer multiple of the wavelength, constructive interference occurs at one detector and destructive interference at the other.
Conversely, a half-wavelength path difference results in reversed interference
conditions.
The interferometer’s ability to precisely measure optical path differences makes
it an ideal instrument for refractive index measurements. By introducing a pressure cell in one arm, variations in air density alter the refractive index, shifting the
interference fringes. The shift in fringe position is directly related to the change
in optical path length, allowing the refractive index to be determined with high
precision.
3
Experimental Procedure
The experimental setup for this study consists of a Mach-Zehnder interferometer configured to observe circular interference fringes and measure the refractive
index of air under varying pressures. The experiment follows a systematic procedure to ensure accurate and reproducible results.
3.1
Setup Alignment and Calibration
1. Align the laser source to ensure a coherent and collimated beam enters the
interferometer. 2. Adjust the first beam splitter to divide the incident laser beam
2
into two coherent paths. 3. Use mirrors to guide both beams along separate optical paths before recombining them at the second beam splitter. 4. Ensure proper
alignment of the second beam splitter to achieve constructive and destructive interference at the output ports. 5. Observe the initial interference pattern on two
different screens, verifying a phase difference of π between them. 6. If necessary,
fine-tune the mirrors to optimize fringe visibility.
3.2
Observation of Circular Fringes
1. Capture interference patterns formed on the observation screens. 2. Adjust
the optical path lengths using micrometer screws to observe changes in fringe
patterns. 3. Record images of the interference fringes, including bright-center
and dark-center configurations.
3.3
Measurement of the Refractive Index of Air
1. Place the pressure cell in one arm of the interferometer. 2. Maintain initial
atmospheric pressure in the cell and note the corresponding fringe pattern. 3.
Gradually reduce the pressure inside the cell using a vacuum pump. 4. Observe
and count the number of fringes that shift due to the change in refractive index.
5. Record pressure values at different stages using a pressure gauge. 6. Calculate the refractive index change using the measured fringe shifts and the known
wavelength of the laser.
An image of the experimental setup is included here to provide a visual representation of the apparatus and its components.
Figure 1: Labeled image of the experimental setup used in this study.
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Figure 2: Images showing the fringes obtained in the experimental setup: (left) with a bright
center and (right) with a dark center.
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Results
The experiment yielded a series of observations regarding the shift in interference
fringes with varying pressure in the pressure cell. The collected data, detailing the
number of fringe shifts corresponding to different pressure values, is presented
in Table 1.
Pressure (mm Hg)
30
70
100
142
170
192
224
250
Fringe Shift
1
2
3
4
5
6
7
8
Table 1: Observed fringe shifts corresponding to different pressure values in the pressure cell.
To visualize the relationship between pressure and fringe shift, a plot of fringe
shift versus pressure was generated. The expected linear trend supports the theoretical prediction that the refractive index change is proportional to pressure
variation.
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Figure 3: Plot of fringe shift as a function of pressure.
To this data, we have fitted a linear equation (y = ax + b), due to the high
value of Pearson’s coefficient of this data. We noted the obtained fitting parameters in Table 2, and a plot of the best-fit line along with the data is shown in
Fig. ??.
Parameters
Slope (a)
Intercept (b)
SSE
R2
Values
0.0319
-0.2005
0.2779
0.9934
Error
0.0011
0.0741
0.0070
0.0231
Table 2: Fitted parameters for pressure vs. fringe shift data.
From the best-fit line we can conclude,
(ni − nf ) =
aλ0
(Pi − Pf )
d
where m is the fitted slope obtained from the data, ni/f is the pressure corresponding to Pi/f , λ0 is the wavelength of the laser light in vacuum, and d is the
length of the vacuum chamber. Assuming n at vacuum to be 1 and the corresponding pressure at vacuum is 0, we obtain n at 1 atm pressure (i.e. 760 mm of
5
Hg) to be
n = 0.0319 × 665 × 10−9 × 760/0.079 + 1 = 1.000204
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Error Analysis
The slope of the above plot is obtained by,
∆m
∆P
a=
where ∆P is the pressure difference corresponding to which the number of fringe
shifts are ∆m. However, the working formula for
nP =
aλ0
P + n0 ,
d
where we have substituted the value of n0 at vacuum (zero pressure), is given by
ln(np −n0 ) = ln(a)+ln(λ0 )−ln(d)+ln(P) = ln(∆m)−ln(∆P)+ln(λ0 )−ln(d)+ln(P).
Thus we obtain the permissible error as,
δ(np − n0 ) δPi ± δPj δd
=
±
.
np − n0
Pi − Pj
d
We note that ∆P is obtained from observing the initial pressure (Pi ) and the final
pressure (Pf ) and taking their difference. We shall also note that n0 does not have
any error. Although δ(∆m) may not be small, we have to ignore it due to the lack
of a scale to measure the shift in the interference pattern. The other term, λ0 , is
substituted from the system (laser) whose error bar is not known. Therefore, the
maximum permissible error is
δnp
|δPi | + |δPj | δd
+
.
=
np − n0 max
|∆P|min
d
From the least count of the pressure gauge we obtain,
δPi = 2 mm Hg
δd = 1 mm
δnp
2|δPi |
δd (2 × 2) mm Hg
1 mm
∴
=
+
=
+
= 0.146
np − 1 max |∆P|min
d
30 mm Hg
79 mm
∴ |δnp | = 5.84 × 10−5
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Deviation from Theoretical Value
The expected theoretical value of refractive index of air is 1. Thus the deviation
from the theoretical value is then given by
Deviation in n =
|1.00028 − 1.00020|
= 7.99 × 10−5
1.00020
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Discussion
6.1
Explanation behind Linear and Circular Fringes
During the experiment, we observed that before placing a lens in one of the arms,
the interference fringes were linear. However, after introducing a convex lens in
one arm, the fringes became concentric circles.
This can be explained as follows: In the first case, the wavefronts of the beams
traveling through both paths remain planar, leading to linear fringes upon interference. However, when a planar wavefront passes through a convex lens, it
transforms into a circular wavefront. Thus, in the second case, one arm retains a
planar wavefront while the other acquires a circular wavefront, and their interference results in concentric circular fringes.
6.2
Phase Acquisition and Interference Patterns
The relative phase difference in the interference pattern between two screens
was found to be π. This is explained using Fresnel’s equations for reflection and
transmission at a dielectric interface. A phase change occurs when light reflects
from a lower to a higher refractive index medium, but not in the reverse case.
Since our dielectric has coating on one side but not on the other, light passing
through the beam splitter acquires an additional phase shift of 2πt/λ, where t is
the optical path length within the beam splitter. The total phase acquired in the
two paths can be expressed as:
2π
(l1 + t)
λ
2π
δ2A = 2π + (l2 + t)
λ
2π
δ2B = π + (l2 + t)
λ
The phase differences at the detectors are:
δ1 = 2π +
2π
(l1 − l2 )
λ
2π
δB = π + (l1 − l2 )
λ
If we maintain l1 = l2 , then:
δA =
7
(1)
(2)
(3)
(4)
(5)
• Detector A: δA = 0 ⇒ Constructive interference
• Detector B: δB = π ⇒ Destructive interference
This confirms that the interference patterns at the two detectors are out of phase
by π.
6.3
Sources of Error
Several factors contributed to experimental uncertainties:
• Misalignment of optical components: Since alignment was done by eye estimation, misalignment of the laser beam height, beam splitters, and mirrors
could have altered the path lengths.
• Sensitivity to mechanical disturbances: The interferometer setup was highly
sensitive to vibrations, which may have affected measurements.
• Manual fringe counting: Fringe shifts were counted manually, introducing
human error in data collection.
• Assumed vacuum conditions: The refractive index of the initial medium was
assumed to be 1, whereas the experiment was conducted in air.
6.4
Error Mitigation Strategies
To improve precision, the following measures can be implemented:
• Using CCD cameras or photodetectors for automated fringe counting and
analysis.
• Employing a fine pressure regulator for precise control over pressure changes.
• Utilizing a well-collimated laser to minimize beam divergence.
• Placing the interferometer on an optical table with vibration damping to
minimize disturbances.
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Conclusion
1. The refractive index of air at 1 atm was determined to be 1.00020 ± 0.146.
2. The interference patterns on the two screens were confirmed to be out of
phase by π.
References
[1] Max Born and Emil Wolf. Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light. Cambridge University Press, 7th
edition, 1999.
[2] Eugene Hecht. Optics. Pearson Education, 5th edition, 2016.
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