Chapter 5. Diffraction Part 2

$: Chapter 5. Diffraction Part 2$
Changhee Lee, SNU, Korea

Optoelectronics EE 430.423.001

2016. 2nd Semester

1/27

2016. 10. 20.

Changhee Lee School of Electrical and Computer Engineering

Seoul National Univ. [email protected]

Chapter 5. Diffraction Part 2



2016. 2nd Semester

2/27

5.5 Fresnel diffraction patterns

hh

RhhRhRhr'r ...)'

11(21')'()( 22/1222/122 ++++=+++=+

[ ]dArnrnrr

eeikUUrrikti

P ∫∫ −−=+−

)',cos(),cos('4

)'(0

π

ω

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

hh

L

LnRLRLR n

1

21

)'

11(

... ,2 ,

−+=

=== λλλ

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

. oft independen ,21

221 nLRRR nn λπππ =−+



2016. 2nd Semester

3/27

5.5 Fresnel diffraction patterns

...321 −+−= UUUU p

The optical disturbance at P is the sum of the contributions from the various Fresnel zones. Since the mean phase changes by exactly 180o from one zone to the next,

1543

3211

21...)

21

21(

)21

21(

21

UUUU

UUUUU

n

p

→++−+

+−+=

∞→

For the case of an infinitely large aperture (no aperture at all), the total optical disturbance at P is ½|U1|.



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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 1st Fresnel zone.



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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.

...531 UUUU p ++=

It is much like a lens, because |Up| is much larger than if there were no aperture. The equivalent focal length is L as given by

λ

21RL =

LR λ=11st Fresnel zone



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Rectangular aperture

yxL

hhr'r

yxR

)(21' 22

222

+++=+

+=

[ ]dArnrnrr

eeikUUrrikti

P ∫∫ −−=+−

)',cos(),cos('4

)'(0

π

ω

Simplifying assumptions: The obliquity factor and 1/rr’ vary so slowly compared to eik(r+r’)/r that they can be taken outside the integral.

dyedxeC

dxdyeCU

x

x

y

y

LikyLikx

x

x

y

y

LyxikP

∫ ∫

∫ ∫

=

= +

2

1

2

1

22

2

1

2

1

22

2/2/

2/)(



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Rectangular aperture

Ly

Lkyv

Lx

Lkxu

λπλπ2 2 ====

kLCUdyedxeUU

u

u

v

v

viuiP

πππ == ∫ ∫ 12/2/

1 ,2

1

2

1

22

)()(0

2/2

siSsCdwes wi +=∫ π

∫∫ ==ss

dwwsSdwwsC0

2

0

2 )2/sin()( ,)2/cos()( ππ

Fresnel integrals

222 )()()( dsdSdC =+

dswsdSdswsdC )2/sin()( ,)2/cos()( 22 ππ ==



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Cornu spiral, a plot of the Fresnel integrals The Cornu spiral is useful for graphical evaluation of the Fresnel integrals. • The limit points s1 and s2 are

marked on the spiral. • A straight line segment drawn

from s1 to s2 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.

∫2

1

2 2/s

s

wi dwe π

222 )()()( dsdSdC =+



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Cornu spiral, a plot of the Fresnel integrals

2)(

2)(

121212

121212

Lyyvvss

Lxxuuss

λ

λ

−=−=−

−=−=−

21 )1(

21)( )( ,

21)( )(

iU

SCSC

+

−=−∞=−∞=∞=∞

[ ] [ ] 2

1

2

1)()()()(

)1( 20 v

vuup viSvCuiSuC

iUU +++

=

For the general case in the normalized form



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10/27

Slits and straightedge

[ ] 2

1)()(

10 v

vp viSvCi

UU ++

=

[ ]

+++

+=+

+= ∞− iviSvC

iUviSvC

iUU v

p 21

21)()(

1)()(

1 2200 2

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



2016. 2nd Semester

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

[ ] 00

2200

21)

21

21(

1

21

21)()(

1)()(

12 Ui

iUUiviSvC

iUviSvC

iUU p

vp =+

+=→

+++

+=+

+= ∞−

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



2016. 2nd Semester

12/27

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, 3rd ed. (McGraw-Hill, 1957)



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13/27

Fresnel diffraction from a slit

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

The diffraction pattern from a slit for different Fresnel numbers NF = a2/λd. corresponding to different distances d from the aperture. • At very small distances (very large

NF), the diffraction pattern is a perfect shadow of the slit.

• As the distance increases (NF decreases), the wave nature of light is exhibited in the form of small oscillations around the edges of the aperture.

• For very small NF, the Fraunhofer pattern is obtained. This is a sinc function with the first zero subtending an angle λ/D = λ/2a.



2016. 2nd Semester

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Fresnel integrals

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

)2/sin()2/cos( 222/2

XiXe Xi πππ +=



2016. 2nd Semester

15/27

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.



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5.6 Applications of the Fourier transform to diffraction Path difference δr

lens theoflength focal ,ˆ

cosinesdirection ,, ,ˆˆˆˆ ,ˆˆ

=+=+=⋅=

=++=+=

LLYy

LXxyxnRr

kjinyjxiR

βαδ

γβαγβα

dxdyedAeU LyYxXikrikP ∫∫∫∫ += /)(~ δ



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

, ,),(),(

),(),(

)(

/)(

LkY

LkXdxdyeyxgU

dxdyeyxgYXU

yxi

LyYxXik

===

=

∫∫

∫∫+

+

νµνµ νµ

Spatial frequency

hygyggyg πννν 2 , ...)2cos()cos()( 002010 =+++=

Diffraction pattern is a Fourier resolution of the aperture function.

5.6 Applications of the Fourier transform to diffraction



2016. 2nd Semester

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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.

)21(

)21sin(

2

2 b

bbdyeU

b

byi

ν

νν == ∫

+

−

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

+−

−=

=

<<−=

∫+

−

bbb

dyebyU

bybbyyg

b

byi

/1

/1)

2cos(

)cos(

2/2/for )/cos()(

2

2

πνπνν

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.



2016. 2nd Semester

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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.

dxdyeyxgU yxi∫∫ += )(),(),( νµνµ

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).



2016. 2nd Semester

20/27

Spatial filtering

),(),(),(' νµνµνµ UTU =

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:

νµνµνµ νµ ddeUTyxg yxi∫ ∫∞

∞−

∞

∞−

+−= )''(),(),()','('

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.



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Spatial filtering

Low-pass spatial filtering



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Spatial filtering

High-pass spatial filtering



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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)( )( yieyg yi φφ +≈=

)()(

)(

)(1()(

21

2/

2/

2/

2/

νν

φ

φν

νν

ν

iUU

dyeyidye

dyeyiUb

b

yib

b

yi

yi

+=

+=

+=

∫∫∫

−−

∞

∞−

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

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



2016. 2nd Semester

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Phase contrast and Phase gratings

)'()'()()()'(' 21'

2'

1 ygygdeUdeUyg yiyi +=+= ∫∫ −− νννν νν

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 , )()( )()( 2121 νννν UUiUU +→+

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 freq.carrier the to90 ofshift phase 0

→



2016. 2nd Semester

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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.

https://en.wikipedia.org/wiki/Holography Recording a hologram Reconstructing a hologram

https://en.wikipedia.org/wiki/File:Holograph-record.svg

https://en.wikipedia.org/wiki/File:Holography-reconstruct.svg

https://en.wikipedia.org/wiki/File:Rainbow_hologram.jpeg



2016. 2nd Semester

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5.7 Reconstruction of the wave front by diffraction, Holography



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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)

Chapter 5. Diffraction Part 2

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Transcript of Chapter 5. Diffraction Part 2