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2 2 exp( jk z)exp[ j z ( f X2 f Y2 )]

2 f X2

2 fY2
)]
Fresnel diffraction between confocal spherical surfaces


Two spherical surfaces are said to be confocal if the center of each lies on the surface of the other. The two spheres are tangent to the planes previously used, with the points of tangency being the points where the z axis pierces those planes.


U ( x, y )
-
U ( , )h( x , y )dd
and
e jkz k 2 h ( x, y ) exp[ j ( x y 2 )] j z 2z
j ( 2 2 ) - j ( x y ) e jkz j 2 z ( x 2 y 2 ) U ( x, y ) e [U ( , )e 2 z ]e z dd j z -

P1
y
r01 P0
x

z

The Huygens-Fresnel principle can be stated as
exp( jkr01 ) 1 U ( P0 ) U ( P1 ) r01 cosds j where is the angle between the outward normal n and the vector r01 pointing from P1 to P0.

1

otherwise
2 2 (x y ) z
Under Fresnel approximation
e H ( f X , f Y ) F {h( x, y )} F { e j z e jkz exp[ j z ( f X2 fY2 )]
jkz j
}
e jkz j z ( x 2 y 2 ) H ( f X , f Y ) F {h( x, y )} F { e } j z e jkz exp[ j z ( f X2 fY2 )]
The Fraunhofer approximation
Under the Fresnel approximation
z ( x ) ( y )
2 2
k
2
- j 2 ( ) j ( 2 2 ) e jkz j 2 z ( x 2 y 2 ) z z 2z U ( x, y ) e [U ( , )e ]e dd j z -


When a wave is not perfectly monochromatic, but it is narrow band, a straightforward generalization of the concept of intensity is given by 2 I ( P) | u ( P, t ) | where the angle brackets signify an infinite time average.
k
x
y
The Fraunhofer approximation requires
k ( 2 2 ) max z 2
e
j
k ( 2 2 ) 2z
1
Thus in the region of Fraunhofer diffraction (or equivalently, in the far field), x y k k jkz - j 2 ( ) j ( x2 y2 ) j ( 2 2 ) e z z U ( x, y ) e 2z [U ( , )e 2 z ]e dd jz -


ຫໍສະໝຸດ Baidu
The quadratic-phase factors in (x,y) and () have been eliminated by moving from the planes to the two spherical caps. The two quadratic phase factors in the earlier expression are in fact simply paraxial representations of spherical phase surfaces, and it is therefore reasonable that moving to the spheres has eliminated them.


One is the approximation inherent in the scalar theory. The second is the assumption that the observation distance is many wavelengths from the aperture, r01>>.
Chapter 4 Fresnel and Fraunhofer Diffraction
Weimin Sun College of Science Harbin Engineering University
Background


The intensity of a scalar monochromatic wave at point P as the squared magnitude of the complex phasor representation U(P) of the disturbance, 2 I ( P) | U ( P) | Note that power density and intensity are not identical, but the latter quantity is directly proportional to the former.

r01 y x
z


r01 z
y
x

If the extent of the spherical caps about the z-axis is small compared with their radii:
r01 z x / z y / z

The Fresnel diffraction equation now becomes x y - j 2 ( ) e jkz z z U ( x, y ) U ( , )e dd j z - which, aside from constant multipliers and scale factors, expressed the field observed on the righthand spherical cap as the Fourier transform of the field on the left-hand spherical cap.
The Fresnel approximation

If z 2 ( x ) 2 ( y ) 2
2 2 2
1 x 2 1 y 2 r01 z ( x ) ( y ) z[1 ( ) ( ) ] 2 z 2 z So
e jkz k U ( x, y ) U ( , ) exp{ j [( x ) 2 ( y ) 2 ]}dd jz - 2z
k

k
k
e e jz

jkz
j
k 2 2 (x y ) 2z
-
[U ( , )e
j
k 2 2 ( ) 2z
]e
- j 2 (
x y ) z z
dd
We recognized (aside from multiplicative factors) to be the Fourier transform of the product of the complex field just of the aperture and a quadratic phase exponential.
x fX z
( f X ) 2 ( f Y ) 2 1 ( f X ) 2 ( f Y ) 2 1 2 2

y fY z
H ( f X , f Y ) exp[ j 2 exp[ j 2
z

z
1 (f X ) 2 ( f Y ) 2 ] (1
The Fresnel approximation and angular spectrum

The transfer function of propagation through free space
f f
2 X 2 Y
z exp[ j 2 1 (f X ) 2 (f Y ) 2 ] H ( f X , fY ) 0


P1

y
r01 P0
x

z
exp( jkr01 ) 1 U ( P0 ) U ( P1 ) r01 cosds j
The term cos is given exactly by


P1
z cos r01

y
r01 P0
x

z



Therefore the Huygens-Fresnel principle can be rewritten exp( jkr01 ) z U ( x, y ) 2 U ( , ) r01 dd j Where the distance r01 is given by r01 z 2 ( x ) 2 ( y ) 2 There are only two approximations in reaching this expression.

where we have incorporated the finite limits of the aperture in the definition of U(,), in accord with the usual assumed boundary conditions.
e jkz k U ( x, y ) U ( , ) exp{ j [( x ) 2 ( y ) 2 ]}dd jz - 2z
The Huygens-Fresnel principle in rectangular coordinate





As shown in figure below, the diffraction aperture is assumed to lie in the (,) plane, and is illuminated in the positive z direction. We will calculate the wavefield across the (x,y) plane, which is parallel to the (,) plane and at normal distance z from it. The z axis pieces both planes at their origins.

In some case, the concept of instantaneous intensity is useful, defined as
I ( P, t ) | u ( P, t ) |

2
When calculating a diffraction pattern, we will generally regard the intensity of the pattern as the quantity we are seeking.