Afectación al Principio de igualdad ante la ley

The exact value of r from figure 3.1 is given by

^ 4 + ( + ( Yo-yif ' (3.6.1.1) This m ay be approximated by

r a z g [ l + l ( 4 ^ ) ^ + 4 ( 4 ^ ) ^ ] . _{Z Zq ^ Zg} (3.6.1.2) and similarly for s we have

z [ l + _{z z z z} ] • (3-6.1.3)

W hen ZQ and z are sufficiently large for the approxim ation to be accurate the point Pg is said to be in the region of Fresnel diffraction. The superposition integral (3.6.4) may be written as a convolution of the input U i(xi,yi) w ith the impulse response. Thus, we have

-i ik z f f ^ ( ( 'z - x i ) ^ + (y2-y|)')

U^(Xyy2) = — G Jju^(X;,y^) e dx^dy,. (3.8.1.4) Expanding the quadratic term we obtain

-i t o f f ^ ( ( ,2 - x y + (y2-y,f)

— e"^ U,(x^,y;)e dx^dy,. (3.8.1.4)

Xz

Expanding the quadratic term we obtain

ik , 2 2, ik , 2 2 ikz e € Xz — ( ^2+72 ) r f — (^1 +72 ) e +ygyi )) e * dxjdy^, (3.6.1.5)

A part from an amplitude term and constant phase factor the integral says th at Ug (xg ,yg ) is the Fourier transform of

Ui(xi,yi) exp( ik (xi2+yi2) / 2 z), (3.6.1.6)

with spatial frequencies u and v given by Xn Yo

u = — , V = — . (3.6.1.7)

Xz Xz

3,gf3 ïtegjQnQfFramhQf^rPififraçtioiL

The Fraunhofer, or far field, approxim ation requires th a t the aperture m ust be small compared w ith the source and observation distances, figure 3.1. We write

k (x^+y^)

Zq>> --- max, (3.6.2.1)

and

k(x^ + yi)

where zqis the source distance and z the observation distance. The

aperture size is given by x and y. The first condition m ay be m et by a collimated beam illuminating the aperture. The second condition determines the minim um observation distance and m ay be satisfied by a lens, used to image the Fraunhofer p attern in its back focal plane. W hen these conditions are satisfied we m ay write equation (3.6.1.5) as

— e e JJU j(X j,yj)e dx^dyj.

(3.6.2.3)

If the aperture function is placed in the front focal plane then we have

This eliminates the quadratic term s in Xg and yg and (3.6.2.3) becomes

2ni , .

r r (%+yzyi)

^ J

®

where f is the focal length and D is a constant. Equation (3.6.2.5) shows th a t the F raunhofer diffraction p a tte rn is the F ourier transform of aperture distribution Uj (x,y) evaluated at frequencies u

a n d V w h e r e

Xg Vo

u , V = — . (3.6.2.6)

a fx

3.7 Fourier optics.

The Fourier transform (FT) occurs frequently in optical signal processing, for example, in the operations of correlation and convolution. It occurs as the Fraunhofer diffraction p attern of an aperture and it relates the optical power spectrum to the visibility envelope of interference fringes (FT spectroscopy). The branch of optics which makes use of the FT is known as Fourier optics.

3.7.1 The phase shifdnsf properties of a lens.

If a wavefront is passed through a lens the lens will introduce a phase delay which is a function of the co-ordinates, (x,y), in the plane of the lens.

A point O, a distance dj from a lens will be imaged a t a point I a distance dg from the lens, figure 3.2. The phase across a spherical wave radiating from O is given by

Uq(r) = (3.7.1.1)

where r is the distance from O. This may be written as

2 2 ,2

,

^ ^ ^ik(x"+y"+drr. (3.7.1.2)

If X and y are much sm aller than dj then by the binomial theorem

Ug(x,y) = e ' . (3.7.1.3)

The lens produces a spherical wave centred on I so the phase across the wave is given by

Uj(x,y) = e . (3.7.1.4)

The phase delay of the lens, (x,y), is the difference betw een (3.7.1.3) and (3.7.1.4). We have

-‘ ■'.(l.i-x z V ) -ik ( d.+d,) 2 d, d,

U,(x,y) = G ' ' e ' ^ . (3.7.1.5) Using the lens equation

1 1 1 _(3.7.1.6)

1 "2

and neglecting the constant phase term we have th a t the phase delay due to a lens of focal length f is given by

U^(x,y) = e ^ ' . (3.7.1.7) 3.7.2 The FouHer transforming properties of a lens.

We wish to determine the effect of a lens on light diffracted by an aperture. Consider the optical system of figure 3.3. The amplitude distribution of light in a plane a distance d^ from a lens is U i(xi,yi), and the distribution a t the lens in plane Pg is Ug (xg ,yg ). The light is transmitted, and propagates to plane Pg w ith distribution Ua (xg ,ya ).

From equation (3.6.1.5), section 3.6.1, the distribution of light at Pg

from an aperture at Pi, putting z = di, is given by

= ^e x p ( i k d j ) e x p ( ^ ( x 2 + y 2 ) ) J|U i(X j,yj)

exp ( 4 " (Xj+yj)) exp ( ( x^x^+y^y^)) dx^dy^. (3.7.2.1) The distribution in plane Pg is multiplied by the effect of the lens and

propagated to plane P g . We have for the effect of the lens —ik 2 2

b^J^2»y2^ = ^(^2*^2^ ( i r r ( x^+y^)), _2f (3.7.2.2.) w here

P(x,y) = 1, inside the lens aperture

= 0, otherwise. (S.7.2.3)

The light distribution, Ug (xg ,yg ), a distance dg beyond the lens a t plane Pg is determined by putting the product Ug (xg ,yg )UL(xg , yg ) into equation (3.8.1.5).

In most remote sensing applications the source will produce waves which are effectively p lan ar a t the sensor. These waves will be focused a t the back focal plane of a lens so we let d g = f. Then Ug (x g ,y g ) may be shown to be given by

UgCxg.Yg) = A exp ( ~ ( x3+y3) ( l ~ ) )

JJUi(Xi,yi) exp ( -2%i ( x^(—) + y^(—) ) dx^dy^ (3.7.2.4)

Xf Xf

where A is a constant. The equation shows th at, bu t for a phase error, the amplitude distribution in plane Pg is the FT of the aperture distribution in plane ?%. The phase error depends on the value of d%. If dj = f then the FT is exact and may be written

Ug(x^,yg) = A JJUj(Xj,yj) exp ( -2%i (x^u + y^v)) dx^dy^, (3.7.2.5) where X. yg u = — and V = — . (3J.2.6) Xf Xf 78