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ECUACIONES DE FRESNEL Coeficientes de reflexión en función del ángulo de incidencia:

R// =

tg 2 (i − r ) tg 2 (i + r )

R↵ =

sen 2 (i − r ) sen 2 (i + r )

E//

E// E↵

E↵ i

i

r

E//

E↵

E//

E//

E↵ E↵ i

i

r

E// E↵

1/7

Ecuaciones de Fresnel

Representación gráfica de los coeficientes de reflexión en función del ángulo de incidencia.

0.8 0.7 0.6

n=1.53

angulo de Brewster = 56.8º

0.5 0.4

R

0.3

R //

0.2 0.1 0.0 0.0

20.0

40.0

i

60.0

80.0

INTENSIDAD DE RAYOS REFLEJADO Y TRANSMITIDO (medios sin pérdidas) Luz reflejada:

I r // = R// I 0 //

I r↵ = R↵ I 0 ↵ I0

Luz transmitida:

I t // = (1 − R// ) I 0 //

I t↵ = (1 − R↵ ) I 0 ↵

Grado de polarización (general)

gP =

I max − I min I max + I min

Para luz reflejada:

g P (reflej ) =

I ↵ − I // I ↵ + I //

i

i

r

Ir

It

Nota: obsérvese que la intensidad I↵ nunca se anula por reflexión porque el coeficiente R↵ nunca es cero. En cambio la intensidad I// se anula cuando R//=0 (ángulo de Brewster).

-2/7-

Ecuaciones de Fresnel PROBLEMA 1. La luz natural incide perpendicularmente sobre el vidrio de una ventana, cuyo índice de refracción es 1.53. ¿Qué porcentaje de la luz incidente es devuelta por la primera reflexión en la superficie y qué porcentaje se transmite al interior? Solución. Calculemos los coeficientes de reflexión para las componentes perpendicular y paralela. Como la luz incide perpendicularmente, el ángulo de incidencia i=0, y el ángulo de reflexión r=0. Por tanto hay que calcular límites de:

sen 2 (i − r ) R↵ = sen 2 (i + r )

Plano de incidencia

tg 2 (i − r ) R// = 2 tg (i + r ) 2

E//

2

 sen(i − r )   sen i cos r − cos i sen r  lim i →0  = lim i →0    = r →0  sen( i + r )  r →0  sen i cos r + cos i sen r  2

2

E↵ 2

Vidrio

 sen r (n − 1)   sen i − sen r   n sen r − sen r   n − 1 lim i →0  = lim r →0  = =    r →0  sen i + sen r   n sen r + sen r   n + 1  sen r (n + 1) 

2

donde se ha aplicado la ley de Snell sen i = n sen r. Como para valores pequeños del ángulo el seno y la tangente tienden al mismo límite, del cálculo anterior podemos concluir: 2

2

 1.53 − 1   n − 1 R↵ = R// =  =  = 0.044   n + 1  1.53 + 1  Puesto que la luz incidente es luz natural, la intensidad en el plano de incidencia y en el plano perpendicular será la misma. Y como los coeficientes de reflexión son iguales entre sí, podemos decir que el porcentaje de luz incidente reflejado es el 4.4%, y el porcentaje transmitido es el 96.6%.

PROBLEMA 2. Un haz plano linealmente polarizado incide sobre una lámina de vidrio de índice de refracción n. El plano de polarización es el plano de incidencia, y el ángulo de incidencia es igual al ángulo de Brewster. ¿Qué porcentaje de la luz incidente se transmite a través de la lámina? SOLUCIÓN: Puesto que el ángulo de incidencia es igual al iB ángulo de Brewster se cumple iB +r=π/2. Además, como el plano de polarización de la luz incidente es 1 paralelo al plano de incidencia, el coeficiente de reflexión paralelo es r i´ tg 2 (i B − r ) tg 2 (i B − r ) = = 0, R1 // = 2 π tg (i B + r ) 2 tg 2

2

es decir, EN EL PUNTO 1 NO HAY RAYO r´ REFLEJADO, y toda la luz entra en la lámina de vidrio. Respecto al punto 2 (salida de la lámina), véase que r=i´, y por lo tanto la ley de Snell nos dice que r´=iB. Por tanto el coeficiente de reflexión en la segunda cara es:

R2 // =

tg 2 (i´− r´) tg 2 (r − i B ) tg 2 (r − i B ) = =0 = tg 2 (i´+ r´) tg 2 (r + i B ) 2 π tg 2

Por lo tanto en el punto 2 tampoco hay rayo reflejado y TODA LA LUZ SE TRANSMITE a través de la cara posterior de la lámina.

-3/7-

Ecuaciones de Fresnel PROBLEMA 3. En el fondo de un recipiente lleno de agua hay una cuña de vidrio. La superficie del agua está iluminada con luz natural que incide según el ángulo de Brewster iB. ¿Qué intensidad de la luz incidente es reflejada directamente por la superficie del agua (rayo reflejado 1)?. ¿Cuál debe ser el ángulo de la cuña para que el rayo que se transmite de nuevo fuera del agua después de reflejarse en la cuña (rayo reflejado 2) esté totalmente polarizado en un plano? Véanse valores de n en el dibujo.

Rayo incidente iB

Rayo reflejado 1 Rayo reflejado 2

aire agua (1.33)

vidrio (1.53)

θ

El ángulo de Brewster en la interfase aire-agua se calcula aplicando la ley de Snell y el hecho de que iB+r=90º. sen i B = n sen r , con r = 90º − i B ⇒ ⇒ sen i B = n sen(90º − i B ) = n cos i B −1

Por lo tanto i B = tg 1.33 = 53º , y el ángulo de refracción es r = 90º − i B = 37 º. Al incidir la luz con el ángulo de Brewster el factor de reflexión R//=0, por lo que toda la luz reflejada que compone el rayo 1 es polarizada perpendicularmente al plano de incidencia y su intensidad es: 2

 sen(i B − r )  2 2 R↵ =   = sen (i B − r ) = sen (53º −37º ) = 0.076 + sen( i r ) B   Así que la intensidad de luz reflejada directamente por la superficie libre del agua es:

I r = I r↵ = I incidente↵ R↵ =

I0 = 0.038 I 0 2

donde I0 es la intensidad incidente y se ha tenido en cuenta que por tratarse de luz natural a cada una de las componentes paralela y perpendicular al plano de incidencia le corresponde una intensidad I0/2. Cálculo del ángulo θ de la cuña. El hecho de que el rayo reflejado 2 al salir del agua esté totalmente polarizado en un plano quiere decir que cuando el rayo incidente entró en el agua y se reflejó en la cuña, incidió sobre esta con el ángulo de Brewster para la interfase agua-vidrio, de modo que el plano de polarización al que se refiere el enunciado ha de ser el plano perpendicular al plano de incidencia (para cualquier otro ángulo distinto del de Brewster en la interfase agua-vidrio la luz reflejada habría tenido componentes en los dos planos). El ángulo de incidencia con que la luz llega a la cuña de vidrio es (véase el dibujo) igual al ángulo de refracción r en la interfase aire-agua (cuyo valor es 37º) más el ángulo de la cuña θ. Ángulo de Brewster para la interfase agua-vidrio: iB ' ' '

nagua sen i B = nvidrio sen(90º − i B ) = nvidrio cos i B

θ

r

nvidrio 1.53 = tg −1 = 49º i = tg 1.33 nagua −1

' B

iB’=r+θ

r

Como i B = r + θ , el ángulo de la cuña es: '

θ = i B' − r = 49º −37º = 12º

θ

-4/7-

Ecuaciones de Fresnel PROBLEMA 4. Un haz de luz natural de intensidad I0 incide perpendicularmente sobre una lámina de vidrio de índice de refracción 1.53. ¿Cuál es la intensidad luminosa reflejada y la intensidad luminosa transmitida por la placa? (Tómense en cuenta las múltiples reflexiones producidas, y compárese el resultado con el del PROBLEMA 1). Para el cálculo detallado de la reflexión y transmisión véase esquema y cálculos en la página siguiente. En nuestro caso la incidencia es normal (θ=0), y entonces 2

2

 n − 1  1.53 − 1  = R = R↵ = R// =   = 0.044   n + 1  1.53 + 1  (véase PROBLEMA 1). Por lo tanto la luz reflejada es: I r = 2 I 0

Y la luz transmitida: I t = I 0

R 0.044 = 2I 0 = 0.084 I 0 1+ R 1 + 0.044

1− R 1 − 0.044 = I0 = 0.916 I 0 1+ R 1 + 0.044

Compárense estos resultados con los del PROBLEMA 1, en que sólo se tenía en cuenta la primera reflexión en la superficie. Evidentemente, al tener en cuenta todas las reflexiones y no solo la primera, el porcentaje reflejado es algo mayor.

-5/7-

Ecuaciones de Fresnel

REFLEXIÓN Y TRANSMISIÓN POR UNA LÁMINA DE CARAS PARALELAS

I0

θ

θ

I0 R(1-R)

I0 R3(1-R)

I0 R(1-R)

I0 R2(1-R)

θ

I0 R3(1-R)2

2

I0 R

I0 (1-R)

θ

θ

I0 (1-R)2

I0 R5(1-R)

I0 R4(1-R)

θ

I0 R2(1-R)2

θ

I0 R4(1-R)2

R puede representar o bien el factor de reflexión R// o bien el factor de reflexión R↵. Intensidad reflejada:

I r = I 0 R + I 0 R(1 − R) 2 + I 0 R 3 (1 − R) 2 + I 0 R 5 (1 − R) 2 + ... = 1 = 1− R2 1 R  1− R = I 0 R + I 0 R(1 − R) 2 = I 0 R 1 + = 2I 0  (1 + R)(1 − R) 1+ R  1+ R

= I 0 R + I 0 R(1 − R) 2 (1 + R 2 + R 4 + ...) = I 0 R + I 0 R(1 − R) 2

Intensidad transmitida:

I t = I 0 (1 − R) 2 + I 0 R 2 (1 − R) 2 + I 0 R 4 (1 − R) 2 + ... = I 0 (1 − R) 2 (1 + R 2 + R 4 + ...) = = I 0 (1 − R) 2

1 1− R = I0 2 1+ R 1− R

(Compruébese que Ir+It=I0). a

Nota: la suma de los infinitos términos de una progresión geométrica de razón r<1 es igual a

-6/7-

1

1− r

Ecuaciones de Fresnel PROBLEMA 6. Sobre el centro de una lámina de vidrio flint (n=1.61) tallada en forma de semicírculo incide un haz de luz natural . Si el grado de polarización de la luz se define como

g P (reflejada ) =

i

I ↵ − I // I ↵ + I //

i

r

determinar la polarización del rayo reflejado y del rayo refractado si el ángulo de incidencia es 30º. Ángulo de incidencia i=30º. El ángulo de refracción se calcula usando la ley de Snell: sen i = n sen r , y por tanto r = arcsen(sen 30º / 1.61) = 18º . Cálculo de los coeficientes de reflexión: 2

tg 2 (i − r ) tg 2 (30º −18º )  tg 12º   = 0.0366 R// = 2 = = tg (i + r ) tg 2 (30º +18º )  tg 48º  2

sen 2 (i − r ) sen 2 (30º −18º )  sen 12º  = =  = 0.0783 sen 2 (i + r ) sen 2 (30º +18º )  sen 48º  La luz incidente (I0) es no polarizada, por lo tanto puede escribirse I 0 // = I 0 ↵ = I 0 / 2 . R↵ =

Luz reflejada:

I r // = R// I 0 // = 0.0366

I0 = 0.0183 I 0 2

I r↵ = R↵ I 0 ↵ = 0.0783

I0 = 0.0392 I 0 2

Luz transmitida:

I t // = (1 − R// ) I 0 // = 0.9634

I0 = 0.4817 I 0 2

I t↵ = (1 − R↵ ) I 0 ↵ = 0.9217

I0 = 0.4609 I 0 2

Grado de polarización Luz reflejada:

g P (reflejada ) =

I r↵ − I r // 0.0392 − 0.0183 = = 0.362 I r↵ + I r // 0.0392 + 0.0183

Luz transmitida:

g P (transmitid a ) =

I t // − I t↵ 0.4817 − 0.4609 = = 0.022 I t↵ + I t // 0.4817 + 0.4609

(Nótese que el grado de polarización para la luz transmitida está definido de modo que gP sea positivo). Observación: el grado de polarización de la luz reflejada es grande en comparación con el de la luz transmitida, que prácticamente no está polarizada. Pregunta: ¿cuál sería el grado de polarización de la luz transmitida si el ángulo de incidencia fuese el ángulo de Brewster?

-7/7-

Problemas interferencias rendijas múltiples (parte I)

1

PROBLEMA 1. Un sistema de doble rendija está formado por dos ranuras muy finas de ancho 2b cuyos centros están separados una distancia 2a, donde 2a es cinco veces mayor que 2b. Determínese cuántas franjas de interferencia aparecerán en el máximo central de difracción proyectado en una pantalla lejana cuando esta doble rendija sea iluminada perpendicularmente con un haz plano de luz monocromática. Solución. La intensidad de difracción en aproximación de Fraunhofer para una rendija múltiple iluminada con luz monocromática es: 2

2

 sen β   sen Nα    I = I 0    β   sen α  donde N es el número de rendijas, I0 es la intensidad de una de ellas,  sen β     β 

2

con β = π

2b sen θ es el término de difración, λ

2

2a  sen Nα  y   con α = π sen θ es el término de interferencia. sen α λ  

θ

Al tratarse de una doble rendija N = 2 y la intensidad queda 2

2

2

 sen β   sen 2α   sen β     cos 2 α I = I 0   = 4 I 0  β sen α β       Según el enunciado 2a=5⋅2b → α=5β, y por lo tanto la intensidad será:

2b

2a 2  sen β   cos 2 5β I = 4 I 0   β  El máximo principal de difracción está comprendido entre los valores -π ≤ β ≤ π. El término de interferencia tiene un máximo cada vez que cos2 5β = 1, es decir, cuando 5β = ±mπ (m=0, ±1, ±2,....). Por tanto los valores de β para los que el término de interferencia tiene máximos en el intervalo -π ≤ β ≤ π son: β = 0, β = ±π/5, β = ±2π/5, β = ±3π/5, β = ±4π/5 y β = ±5π/5 = ±π. Nótese que los máximos extremos β = ±π NO FORMARÁN FRANJAS DE INTERFERENCIA, porque al coincidir con los mínimos de difracción, la intensidad resultante es nula. Por lo tanto el número de franjas que aparece en el máximo principal de difracción es f = 1+4⋅2 = 9.

1.2

m=0 m = +1

1.0

I/4I0 (Intensidad relativa)

m = -1

Mínimo difracción (β = - π)

0.8

m = +2

m = -2

0.6

0.4

m = +3

m = -3 0.2

m = +4

m = -4 0.0 -2

0

β (radianes)

2

Mínimo difracción (β = π)

2

Problemas interferencias rendijas múltiples (parte I)

PROBLEMA 2. Dos rendijas de igual anchura cuyos centros están separados una distancia 2a = 0.20 mm se iluminan perpendicularmente con una onda plana de luz monocromática cuya longitud de onda es λ = 500 nm. Cuando se observa el patrón de interferencia en una pantalla lejana, se comprueba que el máximo de interferencia de orden 5 coincide con el primer mínimo de difracción. Determínese: a) Cuál es la anchura 2b de cada rendija. b) Cuántas franjas de interferencia aparecen dentro del máximo principal de difracción. c) Si la pantalla donde se recoge el patrón de difracción e interferencia está a 2.50 m de la rendija, ¿qué distancia (en mm) hay desde el máximo principal de difracción hasta el primer mínimo, y qué anchura (en mm) tiene cada franja de interferencia, contada desde un mínimo al mínimo adyacente? Solución. Apartado a) Al tratarse de una doble rendija la intensidad del patrón recogido en la pantalla es: 2

2

2

 sen β   sen 2α   sen β     cos 2 α I = I 0   = 4 I 0   β   sen α   β   sen β   donde   β 

2

con β = π

2b sen θ es el término de difración, λ

2

2a  sen 2α  2 y   = cos α, con α = π sen θ es el término de interferencia. λ  sen α 

Llamemos k al cociente entre la separación de los centros y la anchura de una de ellas: k =

2a . 2b

2

 sen β   cos 2 kβ . Según esto, α=kβ, y la intensidad del patrón de difracción será I = 4 I 0  β   2b El primer mínimo de difracción está en β = ±π, por lo tanto β = π sen θ minimo = ± π , y de aquí se deduce λ λ que sen θ minimo = ± . 2b Además, los máximos de interferencia aparecen cuando cos2 kβ = 1, es decir, cuando kβ = ±mπ (m=0, ±1, 2a ±2,....). El máximo de orden 5 (m = ±5) deberá cumplir la condición α = kβ = π sen θ m = 5 = ±5π , es decir, λ λ . debe verificarse que sen θ m = 5 = ±5 2a λ 2a λ En consecuencia, como sen θ minimo = sen θ m = 5 , la relación entre 2a y 2b es ± = ±5 → =k =5 2b 2a 2b

La anchura de cada rendija será por tanto 2b =

2a 0.20 = = 0.04 mm k 5

Apartado b) Dentro del máximo principal de difracción aparece un máximo de interferencia cada vez se cumple la condición cos2 α = 1, y cada máximo corresponde a una franja. Según el enunciado el máximo de interferencia de quinto orden (m = ±5) coincide con el primer mínimo de difracción, y por tanto éste ya no será visible. Así que las franjas visibles serán las que correspondan a α = ±mπ (m=0, ±1, ±2, ±3, ±4). En total, 9 franjas.

3

Problemas interferencias rendijas múltiples (parte I)

Apartado c) 2b sen θ = ± π . Como el ángulo de difracción θ es pequeño se λ verifica que sen θ ≈ tg θ = y/L (véase esquema en esta página), y lo lo tanto la distancia del máximo principal al primer mínimo de difracción es:

El primer mínimo de difracción está en β = π

2b y λL 500 ⋅ 10 −6 ⋅ 2500 = ±π → y = = = 31.25 mm λ L 2b 0.04 (La anchura del máximo principal de difracción es 2y = 62.50 mm) β=π

La anchura de las franjas de interferencia es la misma, cada una de ellas tiene su máximo donde se verifica 2a que cos2 α = 1 y su mínimo donde cos2 α = 0, con α = π sen θ . La distancia que hay entre un máximo y el λ máximo consecutivo es la misma que la que hay entre un mínimo y el mínimo consecutivo, por lo que la 2a 2a z anchura z de una franja se puede calcular haciendo α = π sen θ = π = π , donde la anchura de la λ λ L franja es mucho más pequeña que la distancia a la pantalla. 2a z λL 500 ⋅ 10 −6 ⋅ 2500 =π → z= = = 6.25 mm λ L 2a 0.20 Véase que 9 franjas × 6.25 mm/franja =56.25 mm; el resto, hasta 62.5 mm que es la anchura 2y del máximo principal de difracción corresponde a la “mitad” de cada una de las franjas de quinto orden, m = ±5, que no son visibles por coincidir con el primer mínimo de difracción. π

1.2

m=0

1.0

m = +1

I/4I0 (Intensidad relativa)

m = -1

Mínimo difracción (β = - π)

0.8

m = +2

m = -2

0.6

0.4

m = +3

m = -3 0.2

m = +4

m = -4 0.0 -2

z

z

0

2

β (radianes)

L = 2500 mm

y = 31.25 mm

Mínimo difracción (β = π)

4

Problemas interferencias rendijas múltiples (parte I)

PROBLEMA 3. En el laboratorio se ha medido el patrón de interferencia de una doble rendija (las dos rendijas son idénticas) midiendo la iluminación producida sobre una pantalla situada a (200±1) cm de la doble rendija y utilizando un láser de He-Ne (λ=623.8 nm) para iluminarla. El resultado obtenido se presenta en la figura anexa. Determínese la anchura de cada rendija y la distancia entre sus centros. (Supóngase que las lecturas se hacen en la gráfica con ±0.05 mm).

1.0

Intensidad relativa

0.8

0.6

0.4

0.2

0.0 -3

-2

-1

0

1

2

3

distancia (mm)

Solución. Aparecen 5 franjas de interferencia cuyos máximos están situados en las distancias (medidas desde el máximo central) y = 0, y = ±1 mm, y = ±2 mm. Por tanto deducimos que el siguiente máximo de interferencia que debe estar a y = ±3 mm (los máximos interferenciales de una doble rendija están igualmente espaciados) no aparece por coincidir con el mínimo de difracción. 2

 sen β   cos 2 α La intensidad del patrón está dada por I = 4I 0   β  donde los máximos de interferencia aparecen cuando cos2 α=1, es decir cuando se verifica que 2a α = π sen θ = ± m π (m = 0,±1,±2,...) λ Como la distancia L (=200 cm) entre la pantalla y la doble rendija es grande en comparación con las distancias y, sen θ ≈ tg θ = y/L, y de la posición de los máximos de interferencia podemos obtener la distancia entre centros de rendijas:

2a y λL 632.8 ⋅ 10 −6 ⋅ 2000 632.8 ⋅ 10 −6 ⋅ 2000 = m π → 2a = m = 1⋅ = 2⋅ = 1.2656 mm λ L 1 2 y (se ha escrito con 4 decimales, este número de cifras deberá ajustarse al calcular el error). El ángulo de difracción β está relacionado con la anchura de la rendija; el primer mínimo de difracción corresponde a sen β=0 con β≠0, es decir, el primer mínimo ocurre cuando: 2b 2b ymin β = π sen θ = π = ±π λ λ L Por lo tanto la anchura de la rendija es: 2b y min λL 632.8 ⋅ 10 −6 ⋅ 2000 β=π = π → 2b = = = 0.4219 mm y min λ L 3 α=π

Cálculo de errores. Aceptamos ∆λ=0.1 nm=10-7 mm; ∆y=∆ymin=0.05 mm; ∆L=10 mm. El error máximo cometido en la distancia entre rendijas es (calculamos con m=2, y =2 mm):

Problemas interferencias rendijas múltiples (parte I)

5

 L  ∂ (2a ) λ λL  ∂ (2a )  ∂ (2a ) ∆y  = m  ∆λ + ∆L + 2 ∆y  = ∆L + ∆λ + ∆(2a ) = m  ∂y ∂L y  y  y   ∂λ  2000 − 7 632.8 ⋅ 10 −6  632.8 ⋅ 10 −6 ⋅ 2000 = 2 10 + 0.05 = 0.0001 + 0.0032 + 0.016 ≈ 0.02 mm 10 + 2 2  2  2

El error en la anchura de la rendija es: ∆(2b ) = =

λ λL ∂ (2b ) ∂ (2b ) L ∂ (2b ) ∆y = ∆λ + ∆L + 2 ∆ymin = ∆L + ∆λ + ∂y min ∂L ymin ymin ∂λ y min

2000 − 7 632.8 ⋅ 10 −6 632.8 ⋅ 10 −6 ⋅ 2000 10 + 10 + 0.05 = 0.00007 + 0.0021 + 0.0070 ≈ 0.009 mm 3 3 32

De acuerdo con esto, los valores aceptados para la anchura de la rendija y la distancia entre centros de las mismas son: 2b = (0.422±0.009) mm 2a = (1.27±0.02) mm

1

2

3

4

5

6

7

8

1

2

3

4

DIFRACCIÓN POR UNA RENDIJA (APROXIMACIÓN DE FRAUNHOFER) Rendija de ancho 2b: obtendremos la distribución de intensidad en una pantalla lejana dividiendo la rendija en un gran número de rendijas infinitesimales, cada una de anchura dY. La coordenada Y de cada rendija infinitesimal se mide desde el centro (Y=0) de la rendija real. (Utilizamos aquí el tipo Y mayúscula para distinguir la coordenada de la rendija de la coordenada y en la pantalla donde recogeremos el patrón de difracción). Tomaremos como referencia de desfase nulo el campo eléctrico de la onda que sale del centro de la rendija, y supondremos que es válida la aproximación de Fraunhofer. Suponiendo que cuando la onda emerge de la rendija está uniformemente distribuida a lo largo del eje Y, la amplitud que alcanzará en un punto de la pantalla lejana en la dirección dada por el ángulo θ es E0⋅dY/2b, ya que dY/2b es la fracción de la rendija ocupada por la rendija elemental. (Nótese que decir pantalla lejana implica despreciar cualquier dependencia de la amplitud que llega a la pantalla con el ángulo θ). El campo eléctrico en la pantalla correspondiente a la fracción de la onda que llega desde la rendija elemental mostrada en el dibujo (camino 2) es: dE = E 0

dY j (ωt − kx +ε) e 2b

ε=

θ

Y=b Y=0 dY Y = -b

2π Y ⋅ sen θ λ

Camino 1

(a la pantalla) Camino 2

θ Y

θ Y sen θ

El campo total en el punto de la pantalla indicado por el ángulo θ que es debido a la suma de todos los campos infinitesimales vale: E=

E 0 + b j (ωt − kx + ε) dY ∫e 2b − b

Conviene integrar en función de ε, no en función de Y: cambio de variable y cambio de límites de integración. λε λ Despejando y diferenciando: Y= dY = dε 2π sen θ

2π sen θ

Cuando Y = ±b ⇒ ε = ± (2πb/λ)sen θ. E=

E 0 + b j (ωt − kx + ε) E 0 + ( 2 πa / λ ) sen θ j (ωt − kx + ε) λ e dy = e dε ∫ ∫ 2b − b 2b −( 2 πa / λ ) sen θ 2π sen θ

Llamando β = (2πb/λ)sen θ la integral queda:

1/2

E=

=

+β E 0 + b j (ωt − kx +ε) E0 E 0 +β j (ωt − kx + ε) λ j ( ωt − kx + ε ) ε = e dY = e d dε = ∫ ∫ ∫e 2b − b 2b 2π sen θ −β 2β −β

[

E 0 j (ωt − kx +ε ) e 2β

]

+β −β

=

[

]

(

E 0 j (ωt − kx + β ) E e − e j (ωt − kx − β ) = 0 e j (ωt − kx ) e jβ − e − jβ 2β 2β

)

Para obtener la intensidad en la pantalla multiplicamos por el conjugado complejo: I ∝ E ⋅ E* =

E 0 j (ωt − kx ) jβ E E2 e e − e − jβ ⋅ 0 e − j (ωt − kx ) e − jβ − e jβ = 0 2 2 − e 2 jβ − e − 2 jβ 2β 2β 4β

(

)

(

)

(

)

Según la fórmula de Euler para el seno: e jβ − e − jβ e jβ − e − jβ 2 − e j 2 β − e − j 2 β e jβ − e − jβ sen β = sen 2 β = ⋅ = 2j 2j 2j 4 Por lo tanto la intensidad es: 2

 sen β   = I 0 sinc 2 β I∝E = I 0  2 β  β  2 Donde I0∝E0 y sinc β = (1/β) sen β. La representación gráfica de esta función aparece en la figura siguiente. 2 0

sen 2 β

1.0

Intensidad relativa

0.8

0.6

0.4

0.2

0.0 -10

-8

-6

-4

-2

0

2

4

6

8

10

β (radianes)

MÁXIMOS Y MÍNIMOS. sen β

El máximo principal de la función sinc2 β está en β = 0, ya que lim →1. β→0 β Los mínimos aparecen cuando β = ±mπ (m = 1, 2, 3...), y los máximos secundarios aparecen cuando β = ±1.4303π, ±2.4590π, ±1.4707π..., que son las soluciones de la ecuación trascendente tg β = β (demuéstrese).

Difracción por una rendija (aproximación de Fraunhofer)

2/2

1– Para calcular la expresion que relaciona la distancia focal f’ de la lente y la longitud de onda vamos a usar la ecuación de lentes delgadas: 1 s

1 s

nl

Cuando s

1

1 R1

tenemos

1 R2

s

f

y por tanto 1 f

nl

1 f

1 R1

1

B

A

10 f

A

10 2 f2

A

2

1 R2

1

2 20

B 2

B 2

y finalmente: f’ 2

100 A B2

1

2