Page 184 - Photodetection and Measurement - Maximizing Performance in Optical Systems

Stability and Tempco Issues

Stability and Tempco Issues 177

reflections, and so the visibility is greatly reduced. Calculation of the inter-
ference intensity is straightforward. First we write for the two interfering
amplitudes:

f
A 1 = A e j 1 (8.3)
1
f
A 2 = A e j 2 (8.4)
2
Where |A 1|, |A 2| are the real amplitudes, and f 1, f 2 are the phases of the two
waves. In order to calculate the individual intensities, we multiply the ampli-
tude by its complex conjugate:
is: I 1 = A A 1 * (8.5)
Intensity of A 1 1
2 j 1 - j 1
f
f
= A 1 e (8.6)
2
= A 1 (8.7)
is: I 2 = A A 2* (8.8)
Intensity of A 2 2
2 j 2 - j 2
f
f
= A 2 e (8.9)
2
= A 2 (8.10)
To obtain the combined intensity of the interfering waves, first add the ampli-
tudes to get the total field, then obtain the intensity I as above:
f
f
f
I = ( A e j 1 + A e j 2 )( A e - j 1 + A e - j 2 ) (8.11)
f
1
2
1
2
2 2 j 2 - j 1 j 1 - j 2
f
f
f
f
1
= A 1 + A 2 + A A 2 (e ) + A A 2 (e ) (8.12)
1
2 2 ( j f 2 - 1 ) ( j f 1 - 2 )
f
f
1
1
= A 1 + A 2 + A A 2 (e ) + A A 2 (e ) (8.13)
2 2 jD - jD
1
= A 1 + A 2 + A A 2 (e + e ) (8.14)
where D= f 2 - f 1 is the phase difference between the two waves. Noting that:
e ( jD + e - jD ) = 2 cos D (8.15)
we obtain the resultant intensity:
2 2
I = A 1 + A 2 + 2 A A 2 cos D (8.16)
1
As D varies from 0 to 2p, caused for example by temperature variations in the
beam-splitter material or small changes in incidence angle, the intensity varies
2
2
sinusoidally about a mean intensity of (|A 1| + |A 2| ), with a peak to peak vari-
ation of 4|A 1||A 2|. For example, in the situation of two almost equal-intensity
reflections from a glass slide beam-splitter, the intensity will be 100 percent
modulated. Even if a 1mW signal interferes with only a 4 percent reflection
(0.04mW) from a single glass reflection, the intensity will vary from 0.64 to
Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com)
Copyright © 2004 The McGraw-Hill Companies. All rights reserved.
Any use is subject to the Terms of Use as given at the website.