Chapter 5. Diffraction Part 2

EE 430.423.001 2016. 2^nd Semester

2016. 10. 20.

Changhee Lee

School of Electrical and Computer Engineering Seoul National Univ.

[email protected]

Chapter 5. Diffraction

Part 2

EE 430.423.001 2016. 2^nd Semester

5.5 Fresnel diffraction patterns

h R h

h h R

h R

h r'

r ) ...

' 1 ( 1

2 ' 1 )

' ( )

(

+

+

+

= + +

+ +

= +

[ ^{n r n r} ] ^dA

rr e e

U ikU

r r ik t

i

P

^{= −} ₄

_π

∫∫

_'

^{cos(  ,  ) − cos(  ,  ' )}

Fresnel zones: regions bounded by concentric circles, R=constant, defined such that r+r’ differs by λ /2 from one boundary to the next.

h L h

L n R

L R

L

R

1 2 1

' ) 1 ( 1

...

, 2

, +

=

=

=

= λ λ λ

If R

and R

are the inner and outer radii of the (n+1)st zone, the area is

. of t independen

2

,

1 2

2

1

R R L n

R

π

π λ

π

− =

EE 430.423.001 2016. 2^nd Semester

5.5 Fresnel diffraction patterns

3

...

2

1

− + −

= U U U

U

The optical disturbance at P is the sum of the contributions from the various Fresnel zones.

Since the mean phase changes by exactly 180

from one zone to the next,

1 5

4 3

3 2

1 1

2 ... 1

2 ) 1 2

( 1

2 ) 1 2

( 1 2

1

U U

U U

U U

U U

U

n p



 →

 +

+

− +

+

− +

=

∞

→

For the case of an infinitely large aperture (no aperture at all), the total optical

disturbance at P is ½|U

|.

EE 430.423.001 2016. 2^nd Semester

5.5 Fresnel diffraction patterns

In the case of an irregular obstacle,

(1) If P is in the illuminated region, the presence of the obstacle makes little difference, (2) If it is in the shadow region, the optical disturbance is very small, roughly in agreement

with geometrical optics.

Diffraction fringes appear around the shadow only if the irregularities at the edge of the

obstacle are small compared to the radius of the 1

Fresnel zone.

EE 430.423.001 2016. 2^nd Semester

Zone plate

If an aperture is constructed so as to obstruct alternate Fresnel zones, say the even- numbered ones, then the remaining terms in the summation are all of the same sign.

Such an aperture is called a zone plate.

5

...

3

1

U U

U

U

= + +

It is much like a lens, because |U

| is much larger than if there were no aperture. The equivalent focal length is L as given by

λ

2

R

L =

L

R

= λ

1

Fresnel zone

EE 430.423.001 2016. 2^nd Semester

Rectangular aperture

y L x

h h r' r

y x

R

) 2 (

' 1

2 2

2

+ +

+

= +

+

=

[ ^{n r n r} ] ^dA

rr e e

U ikU

r r ik t

i

P

^{= −} ₄

_π

∫∫

_'

^{cos(  ,  ) − cos(  ,  ' )}

Simplifying assumptions: The obliquity factor and 1/rr’ vary so slowly compared to e

/r that they can be taken outside the integral.

dy e

dx e

C

dxdy e

C U

x x

y y

L iky L

ikx x x

y y

L y x ik P

∫ ∫

∫ ∫

=

=

2

1

2

1

2 2

2

1 2

1

2 2

2 / 2

/

2 / ) (

EE 430.423.001 2016. 2^nd Semester

Rectangular aperture

y L L

y k L v

L x x k

u π λ π λ

2

2 = =

=

=

k L U C

dy e

dx e

U

U

u

v v

v i u

i P

π

π

π

=

=

∫

∫

^,

) ( )

0

(

2

2/

s iS s

C dw

s

e

w

i

= +

∫

∫

∫ ⁼

=

w dw S s

w dw

s

C

2 0

2

/ 2 ) , ( ) sin( / 2 ) cos(

)

( π π

Fresnel integrals

2 2

2

( ) ( )

)

( dC + dS = ds

ds w

s dS ds

w s

dC ( ) = cos( π

/ 2 ) , ( ) = sin( π

/ 2 )

EE 430.423.001 2016. 2^nd Semester

Cornu spiral, a plot of the Fresnel integrals

The Cornu spiral is useful for graphical evaluation of the Fresnel integrals.

• The limit points s

and s

are marked on the spiral.

• A straight line segment drawn

from s

to s

gives the value of the integral

• The length of the line segment is the magnitude of the integral, and the projections on the C and S axes are the real and imaginary parts, respectively.

• ds represents an element of arc.

∫

i

dw

e

2 2

2

( ) ( )

)

( dC + dS = ds

EE 430.423.001 2016. 2^nd Semester

Cornu spiral, a plot of the Fresnel integrals

) 2 (

) 2 (

1 2

1 2 1

2

1 2

1 2

1 2

y L y

v v

s s

x L x

u u

s s

λ λ

−

=

−

=

−

−

=

−

=

−

2 1

( 1 )

2 ) 1

( )

( 2 , ) 1 ( )

(

i U

S C

S C

+

−

=

−∞

=

−∞

=

∞

=

∞

[ ^{( ) ( )} ] [

^{( ) ( )} ]

) 1

(

0 v

v u

u

p

C u iS u C v iS v

i

U U + +

= +

For the general case in the normalized form

EE 430.423.001 2016. 2^nd Semester

Slits and straightedge

[ ^{( ) ( )} ]

1

0 v

v

p

C v iS v

i

U U +

= +

[ ] _ ^{ + + +} _ ^

= + + +

=

C v iS v i

i v U

iS v

i C

U

U

2 1 2 ) 1 ( )

1 ( )

( )

1 (

0

0 ²

F. A. Jenkins and H. E. White, Fundamentals of Optics, 3^rd ed. (McGraw-Hill, 1957)

EE 430.423.001 2016. 2^nd Semester

Straightedge

If the receiving point P is exactly at the geometrical shadow edge, then v₂=0.

[ ]

0

2 ) 1 2 1 2 ( 1 1

2 1 2 ) 1 ( )

1 ( )

( )

1 (

2

i U

i U U

i v

iS v

i C v U

iS v

i C

U

U

+ =

= +

 →

 

 + + +

= + + +

=

The highest irradiance occurs just inside the illuminated region at v₂~1.25, where I_p~1.37I₀.

EE 430.423.001 2016. 2^nd Semester

Narrow slits and opaque narrow strips

Photographs of a number of Fresnel diffraction patterns for single slits of different widths. As the slit becomes wider, the fringes go through very rapid

changes, approaching for a wide slit the general appearance of two opposed straight-edge diffraction patterns.

Fresnel diffraction by narrow opaque strips.

Babinet's principle is not very useful in dealing with Fresnel diffraction. In Fraunhofer diffraction, the diffraction patterns due to complementary screens are identical. In a typical case of Fresnel diffraction, however, this is not true, as may be seen by comparing two Figs.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 3^rd ed. (McGraw-Hill, 1957)

EE 430.423.001 2016. 2^nd Semester

Fresnel diffraction from a slit

The diffraction pattern from a slit for different Fresnel numbers N_F = a²/λd.

corresponding to different distances d from the aperture.

• At very small distances (very large N_F), the diffraction pattern is a perfect shadow of the slit.

• As the distance increases (N_F

decreases), the wave nature of light is exhibited in the form of small oscillations around the edges of the aperture.

• For very small N_F, the Fraunhofer pattern is obtained. This is a sinc function with the first zero

subtending an angle λ/D = λ/2a.

EE 430.423.001 2016. 2^nd Semester

Fresnel integrals

Bahaa E. A. Saleh, Malvin Carl Teich, Fundamentals of Photonics (1991)

) 2 / sin(

) 2 /

cos(

2

2/

X i

X

e

= π + π

EE 430.423.001 2016. 2^nd Semester

5.6 Applications of the Fourier transform to diffraction

Now we consider the general problem of (Fraunhofer) diffraction by an aperture having not only an arbitrary shape, but also an arbitrary transmission including phase retardation, which may vary over different parts of the aperture.

EE 430.423.001 2016. 2^nd Semester

5.6 Applications of the Fourier transform to diffraction

Path difference δr

lens the

of length focal

ˆ ,

cosines direction

, , ˆ , ˆ

ˆ ˆ ˆ , ˆ

= +

= +

=

⋅

=

= +

+

= +

=

L L y Y L

x X y

x n R r

k j

i n y j x i R

β α

δ

γ β α γ β

α





dxdy e

dA e

U

^~ ∫∫

⁼ ∫∫

EE 430.423.001 2016. 2^nd Semester

For a nonuniform aperture we introduce an aperture function g(x,y).

,

, )

, ( )

, (

) , ( )

, (

) (

/ ) (

L kY L

dxdy kX e

y x g U

dxdy e

y x g Y

X U

y x i

L yY xX ik

=

=

=

=

∫∫

∫∫

+ +

ν µ

ν

µ

Spatial frequency

y h g

y g

g y

g ( ) =

+

cos( ν

) +

cos( 2 ν

) + ... , ν

= ² π

Diffraction pattern is a Fourier resolution of the aperture function.

5.6 Applications of the Fourier transform to diffraction

EE 430.423.001 2016. 2^nd Semester

Apodization

Apodization (literally “to remove the feet”) is any process by which the aperture function is altered in such a way as to produce a redistribution of energy in the diffraction pattern. It is an optical

filtering technique, primarily used to remove Airy disks caused by diffraction around an intensity peak, improving the focus.

Consider a single slit. g(y)=1 for –b/2 <y< b/2 and g(y)=0 otherwise.

2 ) ( 1

2 ) sin( 1

2

2

b

b b

dy e

U

b b

y i

ν

ν

= ν

= ∫

Suppose now that aperture function is altered by apodizing in such a way that the resultant aperture transmission is a cosine function:

 

 





− +

= −

=

<

<

−

=

∫

b b

b

dy b e

U y

b y b

b y y

g

b b

y i

/ 1 /

) 1 cos( 2

) cos(

2 / 2

/ for ) / cos(

) (

2 2

π ν π

ν ν

π π

ν

Apodization suppresses the higher spatial frequencies. In this way, it is possible to apodize the circular aperture of a telescope so as to reduce greatly the relative

intensities of the diffraction rings that appear around the images of stars. This enhances the ability of the telescope to resolve the image of a dim star near that of a bright one.

EE 430.423.001 2016. 2^nd Semester

Spatial filtering

The xy plane represents the location of some coherently illuminated object. This object is imaged by an optical system, the image appearing in the x’y’ plane. The diffraction pattern U(µ,ν) of the object function g(x,y) appears in the µν plane.

dxdy e

y x g

U ^{( µ , ν ) =} ∫∫ ^{( , )}

U(µ,ν) is the Fourier transform of g(x,y).

The image function g’(x’,y’) that appears in the x’y’ plane is, in turn, the Fourier transform of U(µ,ν).

EE 430.423.001 2016. 2^nd Semester

Spatial filtering

) , ( ) , ( )

, (

' µ ν T µ ν U µ ν

U =

The finite size of the aperture at the µν plane limits the spatial frequencies that are transmitted by the optical system. And there are lens defects, aberrations, etc., which result in a modification of the function U(µ,ν). All of these effects can be incorporated into the transfer function T(µ,ν) of the optical system, defined as follows:

ν µ ν

µ ν

µ U e

d d T

y x

g ∫ ∫

∞

∞

−

+

= ( , ) ( , )

) ' , ' ( '

The image function g’(x’,y’) is the Fourier transform of the product of T(µ,ν)·U(µ,ν).

The transfer function can be modified by placing various screens and apertures in the µν plane.

This is known as spatial filtering.

EE 430.423.001 2016. 2^nd Semester

Spatial filtering

Low-pass spatial filtering

EE 430.423.001 2016. 2^nd Semester

Spatial filtering

High-pass spatial filtering

EE 430.423.001 2016. 2^nd Semester

Phase contrast and Phase gratings

The method of phase contrast was invented by Zernike, and it is used to render visible a

transparent object whose index of refraction differs slightly from that of a surrounding transparent medium. Phase contrast is particularly useful in microscopy for examination of living organisms.

This method consists of the use of a special type of spatial filter.

) ( 1

)

( y e

i y g =

≈ + φ

) ( )

(

) ( ) ( 1

( )

(

2 1

2 /

2 / 2

/ 2 /

ν ν

φ φ ν

ν ν

ν

iU U

dy e

y i

dy e

dy e

y i U

b b

y b i

b

y i

y i

+

=

+

=

+

=

∫

∫

∫

−

−

∞

∞

−

U₁ and U₂ are 180^o out of phase. The phase-contrast method is inserting a phase plate which shifts the phase of iU₂ by an additional 90^o.

For example, consider a phase grating consisting of alternate strips of high- and low-index material, all strips being perfectly transparent.

EE 430.423.001 2016. 2^nd Semester

Phase contrast and Phase gratings

) ' ( )

' ( )

( )

( )

' (

' y U

e

d U

e

d g

y g

y g = ∫ ^ν

^ν + ∫ ^ν

^ν = +

The phase plate is just a transparent-glass plate having a small section whose optical thickness is λ/4 greater than the remainder of the plate. This thicker section is located in the central part of the µν plane, that is, in the region of low spatial frequencies.

After inserting phase plate ,

U

( ν ) + iU

( ν ) → U

( ν ) + U

( ν )

The g₁ is the image function of the whole object aperture. It represents the constant background.

The g₂ the image function for a regular grating of alternate transparent and opaque strips. Thus, the phase grating has been rendered visible. It appears in the image plane as alternate bright and dark strips.

signal modulated

- amplitude signal

modulated -

phase 

 











 →

EE 430.423.001 2016. 2^nd Semester

5.7 Reconstruction of the wave front by diffraction, Holography

Holography is the science and practice of making holograms. Dennis Gabor was awarded the Nobel Prize in Physics in 1971 "for his invention and development of the holographic method“.

Typically, a hologram is a photographic recording of a light field, rather than of an image formed by a lens, and it is used to display a fully three-dimensional image of the holographed subject, which is seen without the aid of special glasses or other intermediate optics. The hologram is an encoding of the light field as an interference pattern in the photographic medium. When suitably lit, the interference pattern diffracts the light into a

reproduction of the original light field and the objects that were in it appear to still be there, exhibiting visual depth cues such as parallax and perspective that change realistically with any change in the relative position of the observer.

Recording a hologram Reconstructing a hologram

EE 430.423.001 2016. 2^nd Semester

5.7 Reconstruction of the wave front by diffraction, Holography

EE 430.423.001 2016. 2^nd Semester

Homework set #4.

Due date: 2016. 11. 3 (목) Problems in Chapter 5.

1, 4, 8, 11, 15, 19, 20, 21.

Midterm Exam 2

2016. 11. 8(화) 11:00-12:15

시험 범위: Chapter 3~ Chapter 5 (sec. 3.1~3.5 제외)