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(
TEMA - VI
FIBRAS ÓPTICAS.
1. ÁNGULO MÁXIMO DE ACEPTACIÓN.
2. APERTURA NUMÉRICA.
3. CÁLCULO DEL NÚMERO DE REFLEXIONES.
4. CÁLCULO DEL CAMINO ÓPTICO.
5. DISPERSIÓN TEMPORAL.
6. PROBLEMAS RESUELTOS.
7. LISTADO DE PROBLEMAS CON SOLUCIÓN.
FIBRAS ÓPTICAS.
Son hilos cilíndricos de gran longitud y de diámetro
comprendido entre 10 y 80 µm
construídos con materiales dieléctricos. En ellas, la luz que incide sobre una de sus bases se
propaga hasta el otro extremo por sucesivas reflexiones totales.
En sus aplicaciones se utilizan haces de fibras y para
evitar que la luz pase de una fibra a las que le rodean, se
recubren de otro material dieléctrico de menor índice de
refracción con objeto de que se pueda producir la reflexión
total necesaria para la propagación de la luz en su interior,
de manera que:
nf > nC
siendo:
nf = índice de refracción de la fibra.
nC = índice de refracción de su recubrimiento.
1. Ángulo máximo de aceptación.
Si el ángulo de incidencia 1 es
grande (rayo - - de la fig. 2) la luz
llegará a un punto como el P de la
superficie que separa a la fibra de su
recubrimiento con un ángulo
2
pequeño, menor que el ángulo límite
L y el rayo se escaparía de la fibra. Al
disminuir 1 aumenta 2 hasta que
para un cierto valor m del ángulo de
incidencia es 2 = L (rayo - -), de
manera que todos los rayos que
incidieran con ángulos menores que
-) sufrirían reflexión total
m (rayo y se propagarían por el interior de la
fibra.
En consecuencia
Si
Si
1
1
>
m
es el ángulo máximo de aceptación. Resumiendo esta idea:
no hay reflexión total y no hay propagación.
hay reflexión total y propagación de la luz.
m
m
Sea n0 el índice del primer medio. Aplicando la ley de Snell al rayo n0.sen
y como sen m´ = cos
y recordando que sen
m
-:
= nf.sen m´
por ser ambos ángulos complementarios:
L
L
= n´/ n, aplicada a este caso queda: sen
L
= nC / nf
y sustituyendo:
. . . . . . . . . . . . . . . . . (1)
de donde:
. . . . . . . . . . . . . . . . . . . . . . . . . . . (2)
expresión que permite calcular el ángulo máximo de aceptación de una fibra óptica que depende,
únicamente, de los índices de refracción del sistema.
Casos particulares:
a. Si el primer medio es aire (n0 = 1):
. . . . . . . . . . . . . . . . . . . . . . (3)
b. Si, además, la fibra no está recubierta (n0 = nC = 1):
. . . . . . . . . . (4)
y, en este caso, para garantizar que todos los rayos que incidan sobre la fibra se propagen en
su interior, cualesquiera que sea su ángulo de incidencia, debe ser m = 90o y el índice de la fibra
tendría que tener un valor:
2. Apertura numérica de una fibra óptica.
En la expresión (1) el producto n0.sen
m
recibe el nombre de apertura numérica de la fibra:
A.N. = n0.sen
m
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)
Si el primer medio es el aire (n0 = 1), el máximo valor que puede tomar A.N. está condicionado
por el máximo valor que puede tomar el seno de un ángulo, que es la unidad cuando m = 90o ,
en cuyo caso la fibra aceptaría a todos los rayos que llegasen a ella. Si n0 > 1, A.N. puede tomar
valores mayores que la unidad.
3. Cálculo del número de reflexiones.
Por cada reflexión
que sufre, el rayo de luz
recorre un espacio 2y
(fig. 3) avanzando un
espacio 2x, por lo que el
número total de reflexiones
en el interior de la fibra
vendrá dado por la relación
entre la longitud s de la
fibra y lo que avanza en
una reflexión:
N = s / 2x . . . . (6)
y como:
tg ' = ( / 2) / x
con lo que:
siendo
= diámetro de la fibra.
N = (s.tg ') /
2x = / tg '
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)
4. Cálculo del camino óptico.
El camino óptico recorrido por la luz por cada reflexión, de acuerdo con la nomenclatura
empleada en la fig. 3, es:
L1 = nf.2y
y el camino óptico al cabo de las N reflexiones que sufre la luz para recorrer toda la fibra es N
veces el camino óptico relativo a una reflexión:
L = N.L1 = N.nf.2y = (s / 2x).nf.2y = s.nf.y / x = s.nf /cos ´ . . . . . . . . . . . . . (8)
Según se ha visto en el punto 1, la fibra acepta a los rayos que inciden con ángulos
comprendidos entre 0 y m. A estos rayos les corresponde, respectivamente, un camino óptico
mínimo (para = 0) por no sufrir ninguna reflexión y máximo (para = m) por ser el que más
reflexiones sufre:
a.
si
=0
´=0
cos ´ = 1
Lmín = s.nf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)
y según la expresión (8):
b.
si
=
´=
m
' = 90 -
m
cos ´ = sen
L
=
y según (8):
L
= nc / nf
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)
5. Tiempo empleado en recorrer una fibra: dispersión temporal.
El camino óptico expresado en función del tiempo t que tarda la luz en recorrer un medio
es (pág. I. 3) L = c.t siendo c la velocidad de la luz en el vacío, por lo que, sustituyendo el valor
de L dado por la expresión (8), el tiempo empleado por un rayo de luz en recorrer una fibra es:
=
=
ε′
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)
Para una misma longitud de onda, la velocidad con que se propagan los rayos de luz en
el interior de la fibra es la misma, independientemente del ángulo de incidencia. No obstante, el
rayo que tarda menos tiempo en recorrerla es el que incide normalmente ( = 0) por no sufrir
ninguna reflexión interna:
si = 0
´=0
cos ´ = 1
tR = s.nf / c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)
y según (11):
El rayo que emplea más tiempo es el que incide con el ángulo máximo de aceptación
por ser el que más reflexiones sufre en el interior de la fibra:
si
=
m
´=
' = 90 -
m
L
cos ´ = cos m´ = sen
con lo que, según (11), el tiempo empleado por este rayo queda:
=
L
m
= nc / nf
. . . . . . . (13)
La diferencia entre el tiempo empleado por el rayo "más lento" ( tL ) y el empleado por el
"más rápido" ( tR ) es la dispersión temporal :
. . . . . . . . . . . . . . . . . . (14)
( )*+,-./01 *.1 .-2+1
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