Download The Michelson Interferometer and Its Applications

The Michelson Interferometer and Its Applications Interference effects represent an important facet within the subject of optics. In the early 1800s it was Young’s double slit experiment that provided the crucial and unequivocal evidence in support of theories favouring the wave nature of light. Subsequently, by 1865 James Clerk Maxwell had established a comprehensive mathematical theory of electromagnetism. Encapsulated in four simple differential equations, Maxwell’s description of light provided explanation of all empirical observations of the time. Significantly, Maxwell’s formalism showed that light would propagate as a transverse wave with a constant velocity, c. Scientists therefore conjectured the existence of a static “aether,” or a medium pervasive throughout the universe representing an absolute frame of reference, through which light would travel at constant velocity. Though this hypothetical medium would allow for the transmission of electromagnetic radiation, problems arose insofar as the motion of celestial bodies seemed entirely unimpeded. Great effort went into attempts to determine whether observable effects of Earth’s motion and its velocity through the aether could be identified. Enter Albert Michelson. Michelson conjectured that if the speed of light relative to the hypothetical aether was indeed constant, then the Earth’s orbital velocity relative to this aether should exhibit observable relativistic effects. He devised and constructed an optical interferometer with which he presumed he would then be able to detect the relative motion of Earth against the static aether. That is, since Earth’s orbital velocity is approximately 107000 km/h, Michelson anticipated that his interferometer would exhibit measurable change over the course of a year due to the substantial difference in velocity relative to the static background aether. His results, published with Edward Morley in 1887, proved negative. They could identify no observable effects due to the existence of an aether. Consequently, Einstein would soon after publish his theory of special relativity, dispensing with the concept of a static aether entirely and radically pushing physics forward with his hypothesis that the speed of light remained constant in all frames of reference. Interferometry Michelson’s interferometer represented an immensely important tool in as much as it helped to advance our fundamental understanding of physics. However it has proven, along with similar other tools and instruments leveraging interference phenomena, to have wide utility. It allows for extremely precise determination of distances, characterization of spectral bandwidth, the study of the refractive index of gases and transparent media, the characterization of the quality of optical elements, and more. In this laboratory we use the Michelson interferometer to perform a number of experiments to identify the wavelengths of laser and LED light, as well as to study the coherence length of quasimonochromatic sources and broadband (white) light sources. Theory Optical interference results from the interaction of two or more light waves. The pioneering work of Young and his double-­‐slit experiment showed that sufficiently coherent wavefronts from two point sources retain a spatial relationship that allows for constructive or destructive superposition where the fronts overlap. Mathematically, recall that the amplitude of a plane wavefront’s electric field component can be described according to:  

 
E (r, t) = E o cos(k ⋅ r − ω t − θ ) (1) The sum of the electric field components of multiple wavefronts is given, according to superposition, by:   
E = E1 + E 2 +... (2) Since irradiance at any point in space is given by:  2
I= E
T (3)
 2
where E
is the time averaged magnitude of the electric field’s intensity, then in T
the case of two overlapping waves:  2  
 
E = (E1 + E 2 )⋅ (E1 + E 2 ) and it can be shown that: I = I1 + I 2 + I12  2
where I1 = E1
T
 2
, I 2 = E 2
T
(4) (5)  
, and I12 = 2 E1 ⋅ E 2
Further, recall that the third term above, T
.  
I12 = 2 E1 ⋅ E 2
T
(6) represents the interference term for the two waves. From equation (1), it be shown that this expression can be expanded and averaged to give:  
(7) I12 = E o1 ⋅ E o2 cos δ where δ is a phase term dependent on the physical path differences between the 

two waves. Furthermore, taking E o1 and E o2 to be parallel and expressing them in terms of I1 and I2, the total irradiance, I, of the two coherent plane waves can be expressed as: (8) I = I1 + I 2 + 2 I1I 2 cos δ Indeed, the third term in (8) is oscillatory, causing the overall irradiance to vary about the constant terms introduced by I1+I2 as the value of δ ranges between 0 and 2π. Realize though that whereas the treatment provided applies to highly coherent light (well defined phase relationship between the interacting wavefronts), if the sources have insufficient coherence, the poorly defined phase relationship of the overlapping wavefronts causes random superposition, averaging out any maxima or minima in irradiance, leaving only I = I1+I2, such that no interference effect is exhibited. The Michelson Interferometer A version of Michelson’s interferometer is shown in Figure 1. A schematic representation is shown in Figure 2. Figure 1. A Michelson interferometer. (Pasco Scientific) Figure 2. Schematic representation of the Michelson interferometer. (Pasco Scientific) We see from Figure 2 that light from a source strikes a partially silvered beam splitter, causing part of the beam to be transmitted toward a mirror, M1, fixed to a moveable carriage. The remainder of the beam is reflected from the beam splitter toward a fixed mirror, M2. A glass compensation plate along the path toward M1 ensures that the two paths are effectively identical. The two beams, once reflected from M1 and M2, re-­‐converge at the beamsplitter. A precision micrometer screw is connected to the M1 carriage and allows the optical path length along the M1 branch to be altered over a few mm. Fine adjustment screws on M2 allow fine tuning such that the beams from each branch can be overlapped at the viewing screen. When recombined and aligned, the beams fall on the interferometer’s viewing screen where the resultant light exhibits interference effects dependent on the length differences between the two paths. If the distances from the beam splitter to mirrors M1 and M2 differ by distance d, then the total difference in the path lengths traveled by the two beams of light is 2d. When the distance 2d is equal to an integer multiple of the light’s wavelength, constructive interference is observed at the viewing screen as the crests of the two beams overlap, and a bright spot or ring is seen as the viewing screen. Mathematically this condition is described by the equation 2d=nλ (9) where λ is the wavelength of the light and n is an integer. In actuality, due to differences between internal and external reflections at silvered surfaces, a phase reversal of one of beam may result, in which case the equation above becomes appropriate to destructive interference and the condition for constructive interference becomes 2d = (n +1/2)λ (10) Apparatus Pasco Michelson interferometer HeNe laser or diode laser Na light source LEDs (red, green, blue) White light source Screen Diffuser Positive focal length lens Determination of the Wavelength of a Laser 1. Align the HeNe or diode laser beam toward the beam splitter so that the transmitted and reflected beams strike the centers of M1 and M2. Use the fine adjustment screws on M2 to overlap the beams once reflected back to the viewing screen. When done properly, the beam intensity will vary between bright and dark at the output spot when the micrometer screw is slowly turned. By placing a lens or diffuser screen in front of the laser at the interferometer input, the pattern observed at the viewing screen should manifest as concentric light and dark rings (or, due to minor variances in the instrument optics, the pattern may appear more as alternating light and dark lines). Some minor readjustment of the fine alignment screws on M2 may improve the interference NOTE: When the direction of travel of the instrument’s micrometer screw is reversed, a lag in the movement of the M1 carriage will occur. To prevent this from introducing errors when taking micrometer readings, ensure the movement of the screw prior to the reading is in the same direction. That is, do not perform small micrometer reversals when settling into a measurement point. 2. To determine the wavelength of the laser, translate the M2 carriage over some distance, d1-­‐d2, by turning the micrometer screw. Count the number of bright fringes, Δn, observed at the centre of the viewing screen as the micrometer is turned. Many fringes should be observed to improve the accuracy of your measurement. From equation (1), the wavelength of the light is given by 2(d1-­‐d2) = Δnλ (11) 3. Repeat step (2) a number of times to average your result. 4. Compare your result to the accepted value of the wavelength of a HeNe or red diode laser. Measurement of the Wavelength and Coherence Length of an LED Source The coherence length, Lc, of a light is the distance over which successive emissions of radiation from a source maintain a constant phase relationship. The distance is approximated by the relation: Lc ~ λ2/ Δλ (12) where λ is the average wavelength of the spectral line and Δλ is its natural line width, or spectral width (i.e. full width at half maximum). Move mirror M1 to the position where the two arms of the interferometer are of equal length. Determine the distance over which the LED’s interference pattern intensity becomes indistinguishable. 1. As was done for the laser source, determine the wavelength of the LED by counting fringes. 2. Measure the coherence length of the LED source by determining the distance from maximum interference fringe brightness to zero fringe visibility. 3. Use the wavelength and coherence length to calculate the spectral width of the LED and compare to the spectrum provided (see Appendix). 4. Repeat for red, green, and blue LEDs. Would you expect to be able to use this method to determine the coherence length of a HeNe laser? Why or why not? Determination of the Separation of the Wavelengths of a Sodium Doublet The sodium source is dominated by two discrete wavelengths near 590 nm. The beat pattern resulting from the slight difference in the two spectral lines causes a periodic variation in the intensity of the interference fringe pattern. Figure 3. The small difference in the wavelengths of the Na doublet causes periodic variation in the interference fringe intensity. Figure 3 illustrates how the difference in the wavelengths causes variation in the interference fringe pattern intensity. Applying equation (9) at the first position of minimum fringe intensity, d1, gives: 2d1 = m1λ1 = (m1 + n + ½)λ2 (13) where m1 and n are integers. The equation indicates that the condition of minimum fringe intensity occurs when the order of one of the wavelengths of the doublet differs from that of the second wavelength by an odd half-­‐integer. Translating the mirror M1 to the next point of minimum fringe intensity, 2d2 = m2 λ1 = (m2 + n + 3/2)λ2 (14) where m2 is an integer. The difference in the order of the two wavelengths has changed by exactly 1 over the distance of carriage translation, d, from one visibility minimum to the next. Using equations (13) and (14), show that the difference in the wavelengths of the Na doublet is given by λ1-­‐ λ2 = λ2/2d (15) where λ2 ~ λ1λ2 is the square of the average wavelength of the doublet, λ = 589.2 nm. 1. Read the micrometer screw at a point where the two sodium lines exhibit minimum output fringe intensity. Translate M2 to the next region of minimum fringe intensity and read the micrometer screw again. This should occur over a few mm on the micrometer screw. Use equation (15) to calculate the doublet separation. Compare your result to the accepted value recorded in the literature and comment on any discrepancy. Comment on how the resolution of your result compares to that of the Na spectrum provided in the Appendix? Observation of the Interference of White Light Based on your observation of laser and LED interference with the Michelson interferometer, do you expect white light to also exhibit interference? Would you anticipate observing more or fewer interference fringes than for an LED? Move the mirror M1 to the position of equal arm length. Align the white light source to the interferometer. Do you observe fringes? Measure the coherence length of the source by determining the distance on the micrometer over which fringes are visible. If you do observe fringes, what happens if you insert a microscope slide into one branch of the interferometer? Why? Assuming the average wavelength of the source is 650 nm, calculate spectral width of your white light source. APPENDIX – Source Spectra Figure A1. Spectrum of blue superbright LED. Figure A2. Spectrum of green superbright LED. Figure A3. Spectrum of red superbright LED. Figure A4. Spectrum of Na vapour lamp. Figure A5. Spectrum of broadband (white) source. (A Microlab Fibreoptic Source.)