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Multiplying Eq. (8.91) on the left by Eq. (8.92), we obtain:

′ ϕψ ′ = ϕ UU ψ = ϕ ψ (8.93)
−1

which is the result that we are after. In particular, note that the norm of the
vector under this matrix multiplication remains invariant. We will have the
opportunity to study a number of examples of such transformations in
Chapter 9.

8.10.3 Unimodular Matrices
DEFINITION A unimodular matrix has the defining property that its deter-
minant is equal to one. In the remainder of this section, we restrict our discus-
sion to 2 ⊗ 2 unimodular matrices, as these form the tools for the matrix
formulation of ray optics and Gaussian optics, which are two of the major
sub-fields of photonics engineering.

Example 8.16
Find the eigenvalues and eigenvectors of the 2 ⊗ 2 unimodular matrix.

Solution: Let the matrix M be given by the following expression:

 a  b
M =   (8.94)
 c  d

The unimodularity condition is then written as:

−
det(M = ad bc = 1 (8.95)
)
Using Eq. (8.95), the eigenvalues of this matrix are given by:

λ = 1 [(ad ± (ad 2 − 4] (8.96)
+ )
+ )
±
2
Depending on the value of (a + d), these eigenvalues can be parameterized in
a simple expression. We choose, here, the range –2 ≤ (a + d) ≤ 2 for illustrative
purposes. Under this constraint, the following parameterization is conve-
nient:

cos( )θ= 1 ( + ) (8.97)
ad
2

© 2001 by CRC Press LLC