Download Title: Real Fringes in the Michelson Interferometer

Real Fringes in the Michelson Interferometer
Matthew Sinclair
Abstract: Real fringes may be observed through the Michelson Interferometer. 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 order
since the calculations do change. 1
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 of 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. They may be classified into two classes.
The first class is divides the wave-front of a beam, while the second class divides the
amplitude of the beam. The Michelson Interferometer belongs to the second class
mentioned.2 The importance of understanding real fringes in the Michelson comes from
needing to know if the image is real or virtual when making measurements.
Body: To begin with, interferometers work on the basis that light acts as a wave.
Time
varying electric field E and a time varying magnetic field H propagating through space
characterize electromagnetic waves by using Maxwell’s wave equation below in Eq. (1),
where c0 is the speed of light in a vacuum. In Eq. (1), E and H vibrate perpendicular to
one another and perpendicular to the direction of the propagation.3
1 2
 ( E, H )  2 2 ( E, H )
c0 t
2
(1)
Light in a vacuum may be quantified by the equation below where  0 is the free space
permeability and  0 is the free space permittivity.4
c0 
1
(2)
 0 0
This Eq. (2) uses with the characteristics of materials to form the index of refraction, n, as
light propagates through the material by the Eq. (3) where c is the speed light.  r is the
relative permeability, and  r is the relative permittivity when describing index of
refraction in Eq. (4).4
c
n
c0
n
(3)
1
r r
(4)
Sinusoidal waves are the easiest to model as in Eq. (5) using the variables of
amplitude E0, angular frequency   2f (f is the optical frequency), wave number
k
2

, and phase (t  kz   ) (  is the initial phase).3
E ( x, t )  E0 cos(t  kz   )
(5)
Through the description of waves, surfaces of constant phase are considered wavefronts,
while wavefronts 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 may be modeled on a given axis in any directions. Eq. (6) uses a
wavevector k, where r is vector from the origin to point (x,y,z).3
E ( x, y, z, t )  E0 cos(t  k * r   )
(6)
(figure 1)
Using this definition of a plane wave, modeling interference from another plane
wave by the same source will be done by using optical intensity, which is proportional to
2
E0 . To find the optical intensity, the flow of electromagnetic power governed by the
time average of the Poynting vector S=ExH is found.5 From here the optical intensity is
given by the Eq (7).
I  S 
1
c E0
2
2
(7)
Using the definition of E above, the superposition of two waves is as follows in Eq. (811).3
E1  E01 sin( 1t  k1z  1)
(8)
E2  E02 sin( 2t  k2z  2 )
(9)
E  E
1  E 2  E 01 sin( 1t  k1z  1)  E 02 sin( 2t  k2z  2 )

2
2
I  E 02  E 01
 E 02
 2E 01E 02 cos( 2  1 )
(10)
(11)

Eq. (11) 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 (2  1 ) =n*2π when n is a positive
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.3
2
2
I MAX  E02MAX  E01
 E02
 2 E01 E02  ( E01  E02 ) 2  (E0 ) 2
(12)
Now that the fundamentals have been laid down for interfering plain 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.3
(figure 2)
In the illustration above, the following equation is used to calculate the distances
D1, D2, and the wavelength.3
  2  1 
2

( D2  D1 )
(13)
Young’s particular experiment belongs in the first class of interferometers discussed
earlier where wavefronts are split. The variation on this is the second class where the
amplitude splits is the Michelson interferometer. The Michelson interferometer is able to
make precise wavelength measurements, measure very small thicknesses and thicknesses,
and the study of spectral lines.6 The setup of the Michelson interferometer is shown
below.
(figure 3)
Now if E01=E02=E0 and both beams travel the same distance, then the following equation
holds true.
2
2
I  E02  E01
 E02
 2 E01E02 cos(
2

* L  L))
(14)
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. This gives light coming from two point sources
that 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 *  .3
(Figure 4)
Real fringes also may be created through out space with an extended source as shown in
the illustration below.3
(Figure 5)
Here the maximum intensity is at point P on the screen if S1 S1’=n *  and S2’N=n *  .
If n is n-1/2, then Eq. (18) will give an estimate on how far the screen should be away
where b is the thickness of the mirror.3

S2 'S2 S2 'N = /2
(15)
S2 'N  S2 'S2 cos
(16)
S2 'S2  b


(17)
therefore cos  1

2b
(18)
Real fringes may be
seen using a point source or an extended source, while virtual
fringes may only be observed by using an extended source. Extended sources are only
considered relative.1 Therefore, The importance of real and virtual fringes is to note two
particular sets exist.
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 Earthrotated interferometers,” Am. J. Phys. 62, 11 (1994).
3. J. Wilson, J. Hawkes, Optoelectronics: an introduction (Prentice Hall, New York,
1998).
4. E. Hecht, Optics: Fourth Edition (Addison Wesley, San Francisco, 2002).
5. J. Lancis, “High-Contrast White-Light Lau Fringes,” Optics Letters. Vol 29, No. 2
(2004).
6. 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).
7. “Interferometry,” (2005), http://en.wikipedia.org/wiki/interferometry.
8. “Michelson Interferometer,” (2003), http://hyperphysics.phyastr.gsu.edu/hbase/phyopt/michel.html.