Real Fringes in the Michelson Interferometer

Matthew Sinclair

College of Optical Sciences, University of Arizona, Tucson, Arizona 85721

mas13@email.arizona.edu

Abstract:  The difference between real and virtual fringes is described.  A common system that creates real fringes, the Michelson Interferometer, is discussed.

Introduction:  Studies in interferometers grew when Thomas Young performed his double slit experiment, showing that light behaves as a wave instead of a particle1.  Interferometers work on the idea that two coinciding waves with the same phase will amplify each other while two waves having opposite phases will cancel each other out.    From this idea, Young based his experiment.  Interferometers can be classified into two classes.  The first class makes two beams by dividing a common wave-front, while the second class divides the amplitude of the beam.  The Michelson Interferometer belongs to the second class mentioned.2 The Michelson interferometer is able to make precise wavelength measurements, measure very small thicknesses and thicknesses, and the study of spectral lines.3  The importance of understanding real fringes in the Michelson comes from needing to know if the image is real or virtual when making measurements.

 

Theory:  Sinusoidal waves are the easiest to model as in Eq. [1] using the variables of amplitude E0, angular frequency (f is the optical frequency), wave number , and phase  ( is the initial phase).4

                                                                                                                    (1)

Utilizing the definition of waves, surfaces of constant phase are considered wave fronts.  Wave fronts move through space with a velocity v given by v=/k=f, which is called the phase velocity.  This leads into the description of a plane waves.  This is convenient since the plane waves can be modeled on a given axis in any directions.5 

                                                                                 (2)

Figure 1: Vector k projects from the origin to points ky and kx modeling a wave on a given axis.

            Eq. [2] uses a wavevector k, where r is vector from the origin to any point (x,y,z).3  Using the definition of a plane wave, modeling interference from another plane wave by the same source will be done.  This is accomplished by using optical intensity, which is proportional to .  To find the optical intensity, the flow of electromagnetic power governed by the time average of the Poynting vector S=ExH is found.6  From here the optical intensity is given by the Eq [3].

                                                                                                   (3)

Using the definition of E above, the superposition of two waves is as follows in Eq. [4-6].4

                                                                                          (4)

                                                                                        (5)

Omega and the k’s cancel out since the assumption is made that they are the same in both waves.

                                                                                                           (6)

Eq. [6] shows two waves beginning at the same source still having the same angular frequency with a phase difference of , and an interference term now that adds to the total intensity.  This leads to the max intensity for =n2π when n is a positive or negative integer.  The resultant of the two waves is seen as the sum of two irradiances from individual sources plus the interference term.  Therefore, the equation below is said to be coherent, meaning that superposition with interference can be observed.4

                                                 (7)

Now that the fundamentals have been laid down for interfering plane waves from the same source, Young’s double slit experiment proves that light is a wave.  The illustration below shows the fringes created by interference.4 

               

Figure 2:  The diagram above shows how Young’s Double Slit experiment creates fringes.

In the illustration above, the following equation is used to calculate the distances D1, D2, and the wavelength.4 

                                                                              (8)

Young’s particular experiment belongs in the first class of interferometers discussed earlier where wavefronts are split.  The Mickelson interferometer belongs to the second class where the amplitude splits.  The setup of the Michelson interferometer is shown below.

Figure 3: The diffuse monochromatic light source first hits a half silvered mirror acting as a beam splitter.  One beam goes off to a moveable mirror where it is reflected back to the beam splitter where the observer may see it.  The second beam goes through a tilted plane of glass to compensate for the thickness of the beam splitter.  This beam bounces off of a mirror and back to the beam splitter where fringes may be seen from the interference.

Now if E01=E02=E0 and both beams travel the same distance, then the following equation holds true.

                                                           (9)

The Michelson interferometer consists of two mirrors.  One mirror is stationary and the other adjustable.  Both mirrors have tilt and tip adjustments on them.  As shown in the illustration above, the beam first strikes the half-silvered mirror sending 50% through and reflecting 50% to the movable mirror.  The light then propagates back where again, 50% is reflected and 50% goes through to the observer.  This means that only 50% of the light from the source creates the fringes seen.7  The compensating plate is there so both beams pass through the same thickness of glass.  Light from two point sources contribute to one point of light seen by the observer creating a fringe pattern. 

            Real fringes are observed with a point source and an extended source, while virtual fringes require and extended source.8  Below is an illustration of the formation of real fringes for inclined mirrors where S is a real point source, S1 and S2 are virtual point sources, and A and A’ are two half silvered mirrors.  The two beam will interfere constructively if S2N=n.4

           

Figure 4:  Formation of real fringes from a single point source, S.

Real fringes also can be created through out space with an extended source as shown in the illustration below.4 

Figure 5:  Formation of real fringes from an extended light source, S.

Here the maximum intensity is at point P on the screen if S1 S1’=n and S2’N=n. 

                                                                                                      (10)

                                                                                                       (11)

                                                                                                                     (12)

Therefore                                                                                               (13)

 If n is n-1/2, then Eq. [13] will give an estimate on how far the screen should be away where b is the thickness of the mirror.4

Real fringes can be seen using a point source or an extended source, while virtual fringes can only be observed by using an extended source. Extended sources are only considered relative.1  This is a factor when trying to decide how to detect the fringes.  For example, whether to put a CCD array at the output or put a CCD array with a lens to focus the virtual fringes.  Therefore, the importance of real and virtual fringes is to note two particular sets exist.

 

Summary:  Real and virtual fringes both exist when using the Michelson Interferometer.  Knowing when they form is important for detection and measurement.  The use of fringes is important to understand since they may be used to measure wavelength, very fine measurements, and the study of spectral lines.  Although virtual fringes are used for most measurements, it is important to understand when real fringes are obtained in since the calculations do change between real and virtual fringes. 1
Work Cited

1.  A. Zajac, H. Sadowski, S. Light, “Real Fringes in the Sagnac and Michelson Interferometers,” Am. J. Phys. 29, 669 (1961).

2.  R. Anderson, H. R. Bilger, G. E. Stedman, “ ‘Sagnac’ effect: A century of Earth-rotated interferometers,” Am. J. Phys. 62, 11 (1994).

3.  P. T. Ajith Kumar, C. Purushothaman, “A live fringe technique for stress measurements in reflecting thin films, using holography,” Optics & Laser Tech. Vol. 22, No. 1 (1990).

4.  J. Wilson, J. Hawkes,  Optoelectronics: an introduction (Prentice Hall, New York, 1998).

5.  E. Hecht, Optics: Fourth Edition (Addison Wesley, San Francisco, 2002).

6.  J. Lancis, “High-Contrast White-Light Lau Fringes,” Optics Letters. Vol 29, No. 2 (2004).

7.  “Interferometry.” Wikipedia. Nov. 1, 2005.  <http://en.wikipedia.org/wiki/interferometry>.

8.  “Michelson Interferometer.” GSU. Nov. 2, 2003. <http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/michel.html>.