s21_wavemeter

The Measurement of Wavelength for a Monochromatic Source Using Michelson Morley Interferometry

Khalid Al Mahrooqi and Sean Connor

Advisor: Kevin Booth

Introduction

We constructed an apparatus to measure the wavelength of an arbitrary monochromatic light source based upon a modified Michelson Morley interferometer and a piezoelectric-stack mounted mirror in order to accurately determine the wavelength of a given diode laser. In our analysis, we determined the wavelength to be 647.1 nm, less than one standard deviation from the accepted value of 650 nm. We will describe the theory and the detailed construction of the wavemeter, as well as provide our initial results for the helium-neon (HeNe) reference laser and for the two-wave interference of the HeNe laser and the diode laser. We will present two methods of analysis for determining the wavelength of the diode laser; the traditional "fringe counting" technique, as well as computer-assisted Fast Fourier, transform (FFT). We hope that this experiment will be repeated in the future, allowing for the exploration of possible extensions including adapting the wavemeter to accurately measure any arbitrary light source as well as the potential use for atomic and molecular spectroscopy. The two methods are distinct in many ways but the main difference is that the measurement is independent of the mirror displacement, something very difficult to measure accurately. The FFT on the other hand can produce simultaneous results of interfering beams, given that the instrument is calibrated using a known reference monochromatic source.

The Michelson Morley interferometer (henceforth referred to as MMI) was developed in 1887 by Albert A. Michelson and Edward W. Morley in an attempt to detect the Earth's velocity through the supposed luminiferous aether[1]. While the experiment failed to detect such a phenomenon which lead to some of the foundational postulates of Einstein's special theory of relativity, the MMI would prove to be an invaluable apparatus in the area of experimental optics. The device uses a half-silvered mirror in order to split a coherent laser beam into two equal intensity beams. These beams travel along roughly orthogonal paths or "arms" with one arm fixed and the other adjustable. The beams are then reflected back toward the half-silvered mirror and recombined into a single beam which is then measured either by observation or electronically The apparatus operates by adjusting the adjustable mirror's displacement and therefore shifting the relative phase between the two parts of the recombined beam. This leads to a series of bright and dark fringes that can be observed upon a screen. The spacing between the bright or dark fringes can be related to the wavelength of the laser and thus allows for an accurate way to determine said wavelength. Our device operates based upon a modified version of the MMI. We used the Piezo-electric effect on a mirror to displace it at a distance x. Fitted on a piezoelectric stack to control the displacement of the adjustable mirror, as well as a second half-silvered mirror placed directly upstream from the first. This second half-silvered mirror is then used to combine the beams of two laser sources into a single interference beam. By determining the precise pattern of interference, the wavelength of the laser to be measured can easily be computed based upon the known laser's wavelength as well as either the number of fringes observed through a given displacement of the mirror or computationally using the method of FFT. In utilizing two separate methods of measurement, we hope to eliminate or at least minimize error from any single part of the experimental procedure. This allows for a more accurate result in the determination of the given laser's wavelength.

Basic Theory

At the laser source, a coherent monochromatic laser is emitted. This laser is best understood as an electromagnetic wave with intensity described by

The beam is then separated by a half-silvered mirror into two beams with equal frequency and half the original intensity. Each beam is then reflected back along the arm and recombined when it then travels to the detector. The detector receives a combined wave of

with two components E_r and E_m, representing the wave along the fixed arm and the wave along the adjustable arm, respectively. This wave can be described by a path length r and a phase shift

This phase is generally the phase difference between the two waves after recombination, so we are able to set the phase of the fixed arm to be equal to zero. This phase shift manifests as a series of alternating bright and dark fringes if displayed on an observation screen, which in turn can be described as constructive and destructive interference between the two waves. Constructive and destructive interference occurs as the mirror translates and the phase shifts. We divide the wavelength because the displacement changes are experienced by the beam going towards the mirror, and then reflecting away from it, and hence twice the displacement.

respectively, where m is an integer quantifying the number of times the sinusoidal pattern has repeated (fringe count). Therefore, with the displacement known, we can determine the wavelength of the laser by manipulating the equation for constructive interference:

This can be equated to the phase angle as

The detector’s voltage output is proportional to the average intensity of the beam given by

Because the intensity of both beams is equal, the above equation can be simplified by

where I_0 is the maximum intensity. The result is then

Because the maximum intensity is proportional to

we expect to obtain a single, primary peak for each laser after performing the FFT on the associated data. For the case of the two-laser interference beam, we will have a product of two sinusoidal functions, resulting in two distinct peaks from the FFT and one beat frequency resulting from evaluating the intensity which is the integral of the conjugate squared, and therefore the phase difference between the two beams is introduced

Experimental setup

The figure above shows diagrammatically the design for the experimental apparatus. The two lasers are aligned roughly orthogonally and their beams are sent through the first half-silvered mirror in such a way as to combine them into a single beam. This beam then goes through the second half-silvered mirror and is split into two equal beams. These beams then traverse the length of their given arm toward a standard silver mirror upon which they are reflected and made to traverse the arm length a second time. Finally, the half-silvered mirror recombines them into a single beam that is incident upon the silicon-based photodetector. The adjustable mirror is mounted upon a piezoelectric material that is controlled by an AC voltage function generator. The piezoelectric material allows for precise and regular control by means of a changing voltage across it. This causes the material to expand and contract by the phenomenon of piezoelectricity. Through this, the exact velocity of the mirror can be determined by a rough fringe-counting method. The procedure includes:

1. Mirror walking the laser to align the beam parallel to the table using irises and density filters to adjust the intensity

2.Calibrate the Piezo-electric stack mirror displacement using a well-known reference laser, HeNe 632.8nm.

3. Place the second laser into the apparatus as shown above and repeat step 1 before the combination.

4. Block reference laser and record data for fringe counting.

5. Allow for the interference of two beams (beat frequency) and find the wavelength of the unknown laser

Figure: Intensity vs Time graph observed in the oscilloscope. Distortion

in sinusoidal signal is due to the mirror changing direction.

Data and Analysis

The data collected for our experiment is plotted in the figure above. The FFT method yielded a wavelength of 633.5 +/- 10 nm for the HeNe laser and a wavelength of 647.1 +/- 10 nm for the diode laser. The uncertainty in the measurements comes primarily from the resolution of the oscilloscope that was used; the oscilloscope had a sampling frequency of 125 kHz roughly equating to a displacement uncertainty of 10 nm.

The fringe counting method (equation below) yielded similarly accurate results of 642.3 +/- 10 nm for the diode laser. Additionally, the velocity of the adjustable mirror was determined for the purpose of calibration with respect to the piezoelectric stack for the FFT method. This displacement over time t = 0.48 seconds was determined to be 2.28 +/- 0.22 um. Here the fringe number for the HeNe laser and diode lasers are N1 and N2, respectively. If the equation below is applied and the fringes N1 = 69, N2 = 67 (rounded integers of Fraction fringes determined by data analysis), and 632.8 nm(HeNe) for the wavelength, then the resulting wavelength of the diode laser is measured as 651nm.

As observed, both the FFT method and the fringe-counting method yielded results well within one sigma of the expected values of 632.8 nm and 650 nm for the HeNe laser and diode laser, respectively. Some possible errors in these values are due to the condition of the collected data; as seen in the plot above, there appear to be multiple breaks in the sinusoidal pattern for both sets of data. This is the result of the changing in direction of the adjustable mirror when it reaches the limit of the piezoelectric expansion or contraction. This can be further seen in the results for both FFT calculations; there appear to be many significant peaks in both of the plots due to the sinusoidal nature of both data sets not being ideal. However, these FFT plots still show appropriate values for both lasers' wavelengths.

Conclusion

As demonstrated, we used a modified MMI in order to accurately determine the values of the HeNe and diode laser wavelengths within one sigma of the expected values. Further optimization of the experiment by means of improving beam trajectory/alignments and utilizing a more sensitive photodetector and an oscilloscope will lead to even more accurate values for these wavelengths. Allowing for the beams to travel further distances will produce more sensitive instruments, and as a result, more accurate measurements. The Michelson interferometer is very useful in determining very small measurements like gravitational waves in LIGO. To conclude, our results indicate that the fringe counting method and the FFT are useful means of determining the wavelength, in which fringe counting is independent of displacement, and the FFT is simultaneous and contains more information about uncertainties but the accuracy must be improved in order for the results to be conclusive in defining the source.

References

A. Michelson and E. Morley, “On the relative motion of the Earth and the luminiferous ether”, American Journal of Science 34 (203), 333 (1887)
S. P. Davis, M. C. Abrams and J. W. Brault, Fourier transform spectrometry, Academic Press, ISBN-13: 978-0120425105 (2001)
Hecht Optics, 4th edition, Eugene Hecht.
Case Western Reserve University, Encyclopedia of Cleveland History, Michelson Morley experiment.
University physics Volume 3, https://opentextbc.ca/universityphysicsv3openstax/chapter/the-michelson-interferometer/

Page updated

Report abuse