Handbook of Biomedical Optics

Diffraction Optics 21

2.10 Thin Lens

For a thin lens, the eld aer the lens is (Goodman 1968)

U x y U x y P x y ik n x y

2 1

1, , , exp ,

( )

=

( ) ( )

−

(

)

( )

⎡

⎣

⎤

⎦

Δ

(2.82)

where



is the thickness of the glass

n is its refractive index (Figure 2.17)

In the paraxial approximation, x/R

1

≪ 1 and higher powers can

be neglected, so

Δ Δx y

x y

R R

, .

( )

= −

+

−

⎛

⎝

⎜

⎞

⎠

⎟

0

2 2

1 2

2

1 1

(2.83)

us,

U x y U x y P x y ik n

ik n

x y

2 1 0

2 2

1

1

2

, , , exp

exp

( )

=

( ) ( )

−

(

)

⎡

⎣

⎤

⎦

× − −

(

)

+

⎛

⎝

⎜

Δ

⎞⎞

⎠

⎟

−

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

1 1

1 2

R R

.

(2.84)

Putting

1

1

1 1

1 2

f

n

R R

= −

(

)

−

⎛

⎝

⎜

⎞

⎠

⎟

,

(2.85)

we then have

U x y U x y P x y ik n

ik

f

x y

2 1 0

2 2

1

2

, , , exp exp

( )

=

( ) ( )

−

(

)

⎡

⎣

⎤

⎦

− +

( )

⎡

⎣

⎢

⎤

⎦

⎥

Δ

..

(2.86)

e term