Page 122 - Electrical Engineering Dictionary

(2) in optics, region of space that is par- singular or rectangular. A pair (n 1 ,n 2 ) of
tially or totally enclosed by reflecting bound- positive integers n 1 , n 2 such that T pq = 0
aries and that therefore supports oscillation for p< −n 1 and/or q< −n 2 is called the
modes. index of the model. Transition matrices T pq
of the generalized 2-D model satisfy
cavity dumping fast removal of energy
n 1
n 2
stored in a laser cavity by switching the effec- X X
d pq T p−k−1,q−t−1 = 0
tive transmission of an output coupling mir-
ror from a low value to a high value. p=0 q=0
for
cavity lifetime one of several names used
k< 0 and m 1 <k ≤ 2n 1 − 1
to indicate the time after which the energy
t< 0 and m 2 <t ≤ 2n 2 − 1
density of an electromagnetic field distribu-
tion in a passive cavity maybe expected to where d pq are coefficients of the polynomial
fall to 1/e of its initial value; the name pho-
ton lifetime is also common. det [Ez 1 z 2 − A 0 − A 1 z 1 − A 2 z 2 ]
n 1 n 2
X X p q
cavity ratio (CR) a number indicating = d pq z z
1 2
cavity proportions calculated from length, p=0 q=0
width, and height. It is further defined into
ceiling cavity ratio, floor cavity ratio, and and m 1 ,m 2 are defined by the adjoint matrix
room cavity ratio.
adj [Ez 1 z 2 − A 0 − A 1 z 1 − A 2 z 2 ]
cavity short a grounded metal rod con- X X i j
m 1 m 2
necting the body of an RF cavity. By ground- = H ij z z
1 2
ing the cavity, it is kept from resonating. i=0 j=0
(m 1 ≤ n − 1,m 2 ≤ n − 1)
Cayley–Hamilton theorem for 2-D general
model let T pq be transition matrices de-
Cayley–Hamiltontheoremfor2-DRoesser
fined by
model let T ij be transition matrix defined

by
 A 0 T −1,−1 + A 1 T 0,−1 + A 2 T −1,0


 +I n for p = q = 0 

 I (the identity matrix)
ET pq = A 0 T p−1,q−1 + A 1 T p,q−1 

  for i, j = 0


 +A 2 T p−1,q 
 
 A 1 A 2 00
for p 6= 0 and/or q 6= 0
T ij = T 10 := ,T 01 :=
 00 A 3 A 4
and 


 T 10 T i−1,j + T 01 T i,j−1 for i, j ∈ Z +

 0 for i< 0 or/and j< 0
[Ez 1 z 2 − A 0 − A 1 z 1 − A 2 z 2 ] −1
∞ ∞ (Z + is the set of nonnegative integers) of the
X X −(p+1) −(q+1)
= T pq z z 2-D of the Roesser model
1 2
p=−n 1 q=−n 2 " # " #
x h x h
Ex i+1,j+1 = A 0 x ij + A 1 x i+1,j + A 2 x i,j+1 i+1,j = A 1 A 2 ij + B 1 u ij
x v A 3 A 4 x v B 2
+ B 0 u ij + B 1 u i+1,j + B 2 u i,j+1 i,j+1 ij
i, j ∈ Z + where x h ∈ R n 1 and x v ∈ R n 2
ij ij
i, j ∈ Z + (the set of nonnegative integers) are the horizontal and vertical state vectors,
n
where x i,j ∈ R is the semistate vector, respectively, u ij ∈ R m is the input vector,
u i,j ∈ R m is the input, and E, A k , B k and A 1 , A 2 , A 3 , A 4 , B 1 , B 2 are real matri-
(k = 0, 1, 2) real matrices with E possibly ces. The transition matrices T ij satisfy the
c
2000 by CRC Press LLC