Estudios numéricos de esparcimiento inverso de ondas electromagnéticas

CENTRO DE INVESTIGACI ´

ON CIENT´IFICA Y DE EDUCACI ´

ON

SUPERIOR DE ENSENADA

DIVISI ´

ON DE F´ISICA APLICADA

Departamento de ´

Optica

ESTUDIOS NUM´

ERICOS DE ESPARCIMIENTO INVERSO DE ONDAS

ELECTROMAGN ´

ETICAS

TESIS

que para cubrir parcialmente los requisitos necesarios para obtener el grado de

Doctor en Ciencias

presenta:

DEMETRIO MAC´ıAS GUZM ´

AN

RESUMEN de la tesis de DEMETRIO MAC´ıAS GUZM ´

AN, presentada como

requi-sito parcial para la obtenci´

on del grado de DOCTOR EN CIENCIAS en ´

OPTICA,

con orientaci´

on en ´

OPTICA F´ISICA. Ensenada, Baja California, M´

exico, Febrero de

2003

ESTUDIOS NUM´

ERICOS DE ESPARCIMIENTO INVERSO DE ONDAS

ELECTROMAGN ´

ETICAS

Aprobado por:

Dr. Eugenio R. M´

endez M´

endez,

Director de Tesis

El tema central de esta tesis es el desarrollo de algoritmos num´

ericos para la

re-construcci´

on del perfil de superficies rugosas, utilizando datos de esparcimiento en el

campo lejano. Espec´ıficamente, se han propuesto dos algoritmos de inversi´

on, ambos

con algunas variantes. El primero de ellos est´

a basado en principios de

empatamien-to de frentes de onda y hace uso de la informaci´

on de amplitud y fase del campo

esparcido por la superficie. El otro algoritmo utiliza la intensidad esparcida en el

campo lejano y aborda el problema de esparcimiento inverso como un problema de

optimizaci´

on no lineal. Los datos de entrada para ambos algoritmos fueron generados

en forma num´

erica utilizando un m´

etodo riguroso basado en el teorema integral de

Green.

Palabras clave: esparcimiento de luz, esparcimiento inverso, microscop´ıa confocal,

ABSTRACT of the Thesis of DEMETRIO MAC´ıAS GUZM ´

AN, presented in partial

fulfilment of the requirements for the degree of DOCTOR IN SCIENCES in OPTICS,

with major in PHYSICAL OPTICS. Ensenada, Baja California, Mexico, Febrero de

2003

ESTUDIOS NUM´

ERICOS DE ESPARCIMIENTO INVERSO DE ONDAS

ELECTROMAGN ´

ETICAS

Abstract approved by:

Dr. Eugenio R. M´

endez M´

endez,

Thesis advisor

The subject of this thesis is the development of numerical algorithms for the

recons-truction of the profiles of one-dimensional rough surfaces using far-field scattered

data. Specifically two inversion algorithms have been proposed, one of them is based

on wave-front matching principles and uses information of the amplitude and phase

of the scattered field. The other algorithm uses far-field scattered intensity and

ap-proaches the inverse problem as a non linear optimization problem. The input data

for the two algorithms was generated numerically using a rigorous method based on

Green’s Integral Theorem.

Key words: Light scattering, inverse scattering, confocal microscopy, interferometry,

Este trabajo est´

a especialmente dedicado

A

Gabriela (Gaviota) Fumagalli

, por todo lo maravilloso que su llegada ha tra´ıdo

consigo.

A mi madre

Luz Margarita Guzm´

an

y a mis hermanos

Juan

,

Paloma

,

Eugenia

y

Gabriela

, quienes a pesar de la distancia siempre han estado presentes.

A la memoria de mi padre

Eugenio Mac´ıas

.

Agradecimientos

No solamente por sus valiosas cr´ıticas y observaciones durante el desarrollo de este

trabajo, sino tambi´

en por sus ense˜

nanzas a lo largo de mi estancia en este centro

de investigaci´

on, quiero expresar mi m´

as sincero agradecimiento a los miembros de

mi comit´

e de tesis,

Dr. Eugenio R. M´

endez M., Dr. Neil C. Bruce, Dr. Anatolii

Khomenko F., Dr. Gustavo Olague y Dr. V´ıctor Ru´ız

.

A

Ileana y a Claudio

por su amistad, por los interminables ”recreos” y tambi´

en

por los asados.

A

Blanca

por el privilegio de su amistad.

A las secretarias de la jefatura

Olga, Ana

y

Carmen

, mil gracias por todo!

A todos aquellos que en alg´

un momento compartieron conmigo esta experiencia.

Al Consejo Nacional de Ciencia y Tecnolog´ıa (CONACYT) el apoyo econ´

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|

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ζ

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x

1

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|

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x

3

=

ζ

(

x

1

)

*

ψ

R

(

x

1

, x

3

) =

_∞

−∞

dq

2

π

R

(

q

|

k

) exp

i

[

qx

1

+

α

0

(

q

)

x

3

]

,

93@:

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x

3

=

ζ

max

(

x

1

)

0

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ψ

₀

exp

{

ikx

₁

−

iα

₀

(

k

)

ζ

(

x

₁

)

}

=

−

_∞

−∞

dq

2

π

R

(

q

|

k

) exp

i

[

qx

1

+

α

0

(

q

)

ζ

(

x

1

)]

9D2:

% $ !" $ "5 <$ $ %5 "% $ %5

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# $ A$ > " $0 "$$ $ "5 9D2:

exp

{−

ipx

₁

}

& %

x

1

* % $$& '

ψ

₀

_∞

−∞

dx

₁

exp

{

i

(

k

−

p

)

x

₁

−

iα

₀

(

k

)

ζ

(

x

₁

)

}

9D4:

=

−

_∞

−∞

dx

₁

_∞

−∞

dq

2

π

R

(

q

|

k

) exp

i

[(

q

−

p

)

x

1

+

α

0

(

q

)

ζ

(

x

1

)]

.

9D1:

A

I

(

γ

|

Q

) =

_∞

−∞

dx

₁

exp

{−

i

[

Qx

₁

+

γζ

(

x

₁

)]

}

9D3:

%$$& $<%5 ""5 " >$&7" " -'

ψ

₀

[

I

(

α

₀

(

k

)

|

p

−

k

)] =

−

_∞

−∞

dq

$B " /%5 % >$$ *

exp

−

iγζ

(

x

₁

) = 1 +

−

iγ

1!

ζ

(

x

1

) +

(

−

iγ

)

2

2!

ζ

2

(

x

1

) +

. . . ,

9D?:

$ -"5

I

(

γ

|

Q

)

!"

I

(

γ

|

Q

) =

_∞

−∞

dx

₁

exp

{−

iQx

₁

}

1 +

−

_1!

iγ

ζ

(

x

₁

) +

(

−

iγ

)

2

2!

ζ

2

(

x

1

) +

. . .

= 2

πδ

(

Q

) +

−

iγ

1!

ζ

ˆ

(1)

(

Q

) +

(

−

iγ

)

2

2!

ζ

ˆ

(2)

(

Q

) +

. . .

,

9DF:

ζ

ˆ

n

₍

_Q

₎

% $ %- "

ζ

n

₍

_x

1

)

0 "%>" $

"5 9DD: %#

−

ψ

₀

2

πδ

(

p

−

k

) +

−

iγ

1!

ζ

ˆ

(1)

(

p

−

k

) +

(

−

iγ

)

2

2!

ζ

ˆ

(2)

(

p

−

k

) +

. . .

=

−

_∞

−∞

dq

2

π

R

(0)

_s

(

q

|

k

) +

R

(1)

_s

(

q

|

k

) +

R

(2)

_s

(

q

|

k

)

×

×

2

πδ

(

p

−

q

) +

−

iγ

1!

ζ

ˆ

(1)

(

p

−

q

) +

(

−

iγ

)

2

2!

ζ

ˆ

(2)

(

p

−

q

) +

. . .

,

9DC:

R

(

n

)

s

(

q

|

k

)

% $ #"5 $ $" %

ζ

n

(

x

₁

)

0

$*%".% &"$$%.%%,% ><$%%

$%

R

s

(

q

|

k

) =

ψ

0

−

2

πδ

(

p

−

k

) + 2

iα

₀

(

k

)ˆ

ζ

(1)

(

p

−

k

) +

· · ·

9DE:

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$%"A0

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<%5 $ 0

# % & '(

$ '

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