Interferometer Experiment Lab Report

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Abstract

This report presents an experiment using an interferometer to measure the thickness of a nanometer-scale bubble film. The interference phenomenon is employed to extract information, and a Mach-Zehnder interferometer is used in this study. We measured the thickness at a 90-degree position of four bubbles created from different concentrations of solutions, with all results on the scale of 1000nm. During the measurement, we observed fringe patterns, mainly the Fizeau fringe pattern, which are analyzed to determine the fringe shift. The results are then related to the concentration of the bubble solution and the color of the bubbles.

Introduction

This experiment aims to measure the thickness of a bubble film on a nanometer scale using an interferometer.

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Interferometers are powerful tools that exploit interference patterns to extract information with high accuracy and resolution.

Introduction to Interferometer

Interferometers are versatile instruments used in various scientific and engineering disciplines. They function by utilizing interference patterns generated by superimposed waves. In this experiment, we employ a Mach-Zehnder interferometer.

In the Mach-Zehnder interferometer setup, a laser beam is expanded in diameter and then split by a beam splitter.

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Half of the light continues along its original path, while the other half is reflected at an angle equal to the angle of incidence. The separated laser beams are recombined at a second beam splitter, with their diameter reduced by a reducing telescope. This setup allows the laser beam to be captured by a CCD camera's chip, forming an image on a screen.

When an object is introduced into one of the laser paths, it causes a difference in the path lengths of the two laser beams.

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This object will have a refractive index different from that of air, resulting in a phase change in the obstructed path. Since the interference pattern at the camera depends on the relative phase of the beams, this phase change leads to a change in the interference pattern, causing the fringes to move in a direction perpendicular to their length. This phenomenon is known as "fringe shift."

Background: Two Types of Fringes

There are two types of fringes observed in interferometry:

Fizeau Fringes: These fringes are formed by perfectly collimated but not perfectly overlapped beams. They result from a small difference in the path lengths of the beams.
Haidinger Fringes: These fringes are formed by the interference of monochromatic and coherent beams that are not perfectly collimated to each other. The beams slowly diverge over long distances, causing a difference in their path lengths, resulting in fringes that emerge radially outwards.

Theory

For a laser traveling in an empty path, the phase change in air (Ï_air) is given by:

Ï_air = 2ÏxN_airÎ» â 2ÏxÎ»

Where:

Ï_air is the phase change of the light wave in air
x is the difference in path (twice the thickness of the bubble film)
N_air is the refractive index of air (approximated to 1)
Î» is the wavelength of the laser beam

For a laser traveling in the path where the bubble is placed, the phase change in the object (bubble) is given by:

Ï_object = 2ÏxN_objectÎ»

Where:

Ï_object is the phase change of the light wave in the object (bubble)
N_object is the refractive index of the object (bubble)

The change in phase difference between the two lasers is calculated as:

Î´Ï = 2Ïx(N_object - N_air)Î» â 2Ïx(N_object - 1)

For a fringe shift (F) of 1, a dark fringe shifts to the neighboring dark fringe, causing a phase difference change of 2Ï (Î´Ï = 2Ï). Therefore, we have:

Î´Ï = 2ÏF

To obtain the thickness (t) of the bubble film (which is half of x), we use the equations:

t = x/2 = FÎ»/2(N_object - 1)

Experimental Methods

Setup, Alignments, and Preparations

The experimental apparatus is set up as shown in Figure 1, with the telescope part configured as in Figure 3. Initial alignment steps involve ensuring that both laser beams are centered at beam splitter 2 by adjusting the mirrors. A white card is placed at the focal point in the telescope to create two collimated beams. Beam splitter 2 is adjusted so that the two spots on the white card overlap, ensuring that the beams enter the lens in a parallel fashion. By adjusting the turning mirror, the beams are centered before reaching the camera. A white card is used to monitor the beam positions throughout the experiment.

Subsequent experiments involve making small adjustments to the beamsplitter to observe the fringe patterns. The fringe pattern observed is a combination of Fizeau fringes and Haidinger fringes, resulting from small deviations in beam alignment. The goal is to achieve a relatively straight and wide fringe pattern, suitable for observing fringe shifts without excessive blurring.

Measurement of Bubble Film Thickness

For the experiment, four bubble solutions with different concentrations are prepared. Bubbles are blown and carefully positioned between the probe mirror and beam splitter 2. The size and position of the bubble should allow its image to be observed in the center of the displayed image and at the top (90-degree position) of the bubble. The original fringe width is set relatively wide to facilitate clear observations of fringe shifts. Videos of the bubbles are recorded using a CCD camera.

Data Analysis and Results

Two images, one just before and one just after the bubble burst, are selected from each video. Image analysis is performed using software such as ImageJ to determine the shift of a specific fringe at the top (90-degree position) of the bubble. The refractive index of the bubble is assumed to be the same as that of water (1.33), and the wavelength of the laser beam is 550nm (green). The thickness of the bubble film (t) is calculated using the equation:

t = x/2 = FÎ»/2(N_object - 1)

The results indicate that the measured thickness of the bubble film is generally greater than 1500nm. By analyzing the relationship between bubble color and thickness, we deduce that the color of the bubble at its top should be relatively dull. Furthermore, the results suggest that, in general, as the concentration of the bubble solution increases, the thickness of the bubble film also increases, aligning with our expectations.

Conclusion

In this interferometer experiment, we successfully measured the thickness of a nanometer-scale bubble film. The results indicate variations in thickness related to bubble color and solution concentration. This demonstrates the effectiveness of interferometry in accurately determining nanoscale dimensions and properties.

Appendix A: Interferometry Derivation

The mathematical formulation for interferometry is derived as follows:

Er(x, y) = Er.0(x, y)exp(i(wt - kr Â· x))

Ep(x, y) = Ep.0(x, y)exp(i(wt - kp Â· x))

ET ot = Er Ep

IT ot = ((|Er,0|)² (|Ep,0|)²)(1 - 2|Er,0Â·Ep,0| |Er,0|² |Ep,0|² cos(Ï(x, y) Î´Ï(x, y)²)

Above is the mathematical formulation for interferometry, where Î´Ï(x, y) represents the phase difference due to fringe shift of a certain fringe chosen from the whole image in our experiment.