chapter 5. diffraction part 1 - seoul national...

$: Chapter 5. Diffraction Part 1 - Seoul National Universityocw.snu.ac.kr/sites/default/files/NOTE/Lecture (4)_0.pdfthe region of geometrical shadow of the obstacle. • The essential$
Changhee Lee, SNU, Korea

Optoelectronics EE 430.423.001

2016. 2nd Semester

1/27

2016. 10. 18.

Changhee Lee School of Electrical and Computer Engineering

Seoul National Univ. [email protected]

Chapter 5. Diffraction Part 1



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• Diffraction is defined as the bending of light around the corners of an obstacle or aperture into the region of geometrical shadow of the obstacle.

• The essential features of diffraction can be explained qualitatively by Huygens’ principle. The Huygens’ principle states that every point on a wavefront actd as the source of a secondary wave that spreads out in all directions. The envelope of all the secondary waves is the new wave front. Augustin Jean Fresnel (1788-1827) in 1818 explained the diffraction phenomena using the Huygens’ principle and Young’s principle of interference Huygens-Fresnel principle

• We use a more quantitative approach, the Fresnel-Kirchhoff formula to various cases of diffraction of light by obstacles and apertures.

5.1 General description of diffraction

https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle

Diffraction of a plane wave at a slit whose width is several times the wavelength.

Diffraction of a plane wave when the slit width equals the wavelength



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Green’s theorem

)()()( ,F

,FF theoremDivergence

)()(

2

22

VUVUVUUVVU

dVdA

dVUVUVdAVUUV

n

nn

∇⋅∇+∇=∇⋅∇∇−∇=

⋅∇=∇

∇−∇=∇−∇

∫∫∫∫∫∫∫∫∫∫

5.2 Fundamental theory

2

2

22

2

2

22

1

1

tV

uV

tU

uU

∂∂

=∇

∂∂

=∇

0)( =∇−∇∫∫ dAVUUV nn

If both U and V are wave functions and have a harmonic time dependence of the form eiωt.



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5.2 Fundamental theory

reVV

tkri )(

0

ω+

=Suppose that we take V to be the wave function

0)()( 2 =Ω∂∂

−∂∂

−∇−∇ ∫∫∫∫ = dr

er

UrU

redA

reUU

re

r

ikrikrikr

nn

ikr

ρρ

Since V becomes infinite at P, we must exclude that point from the integration. Subtract an integral over a small sphere of radius r=ρ centered at P and then let ρ shrink to zero.

PP UdU π4=Ω∫∫

dAUr

er

eUU n

ikrikr

nP ∫∫ ∇−∇−= )(41π

Kirchhoff integral theorem

U = optical disturbance



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Fresnel-Kirchhoff formula Determine optical disturbance reaching the receiving point P from the source S. V

Two basic simplifying assumptions: (1) The wave function U and its gradient contribute negligible amounts to the integral

except at the aperture opening itself. (2) The values of U and grad U at the aperture are the same as they would be in the

absence of the partition.

'

)'(

0 reUU

tkri ω−

=The wave function U at the aperture

−=

∂∂

=∇

−=

∂∂

=∇

2

''''

2

'')',cos(

'')',cos(

'

),cos(),cos(

re

rikern

re

rrn

re

re

rikern

re

rrn

re

ikrikrikrikr

n

ikrikrikrikr

n

dAr

er

er

er

eeUUikr

n

ikrikr

n

ikrti

P ∫∫ ∇−∇=−

)''

(4

''0

π

ω

Smaller than the 1st term if r, r’>> λ.

Smaller than the 1st term if r, r’>> λ.



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Fresnel-Kirchhoff formula

[ ]dArnrnrr

eeikUUrrikti

P ∫∫ −−=+−

)',cos(),cos('4

)'(0

π

ω

[ ]

'

,1),cos(4

'0

)(

reUU

dArnr

eUikU

ikr

A

tikriA

P

=

+−= ∫∫− ω

π

Fresnel-Kirchhoff integral formula

Circular aperture, 1)',cos( constant,' −== rnr

=obliquity factor )]',cos(),[cos( rnrn −



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Complementary apertures. Babinet’s principle If the aperture is divided into two portions A1 and A2 such that A= A1 + A2. The two apertures A1 and A2 are said to be complementary. From the Fresnel-Kirchhoff integral formula, UP= U1P + U2P (Babinet’s principle)

If UP=0, U1P = - U2P The complementary apertures yield identical optical disturbances, except that they differ in phase by 180o. The intensity at P is therefore the same for the two apertures.



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Babinet’s principle

http://userdisk.webry.biglobe.ne.jp/006/095/15/N000/000/004/136844068624713202721.JPG



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Babinet’s principle



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5.3 Fraunhofer and Fresnel Diffraction

Fraunhofer diffraction occurs when both the incident and diffracted waves are effectively plane. This will be the case when the distances from the source to the diffracting aperture and from the aperture to the receiving point are both large enough for the curvatures of the incident and diffracted waves to be neglected. If either the source or the receiving point is close enough to the diffracting aperture so that the curvature of the wave front is significant, then one has Fresnel Diffraction.



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5.3 Fraunhofer and Fresnel Diffraction

...)1'

1(21)

''(

'')()'('

2

22222222

++++=

+−+−+++++=∆

δδ

δδ

dddh

dh

hdhdhdhd

The quadratic term is a measure of the curvature of the wave front. The wave is effectively plane over the aperture if

λδ <<+ 2)1'

1(21

ddCriterion for Fraunhofer diffraction

The variation of the quantity r+r’ from one edge of the aperture to the other is given by



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5.4 Fraunhofer Diffraction Patterns

[ ]dArnrnrr

eeikUUrrikti

P ∫∫ −−=+−

)',cos(),cos('4

)'(0

π

ω

Simplifying assumptions: (1) The angular spread of the diffracted light is small enough for the obliquity factor not

to vary appreciably over the aperture and to be taken outside the integral. (2) eikr’/r’ is nearly constant and can be taken outside the integral. (3) The variation of eikr/r over the aperture comes principally from the exponential part,

so that the factor 1/r can be replaced by its mean value and taken outside the integral.

dAeCU ikrP ∫∫=



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5.4 Fraunhofer diffraction patterns for the single slit

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



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0for of valuethe

sin

0

0

==

+=

yrr

yrr θFor a single slit of length L and width b, dA=Ldy.

CbLeCkb

C

k

kbLCe

LdyeCeU

ikr

ikr

b

bikyikr

0

0

0

' ,sin21

)sin('

sin

)sin21sin(

2

2

2

sin

==

=

=

= ∫+

−

θβ

ββ

θ

θ

θ



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5.4 Fraunhofer diffraction patterns for the single slit The irradiance distribution in the focal plane is

The maximum value occurs at θ=0, and minimum

values occur for β=mπ=±π, ±2π, ±3π, …

The 1st minimum, β=π, sinθ=2π/kb=λ/b.

slit theof area 0,for irradiance

)sin(

20

20

2

∝==

==

CbLI

IUI

θ

ββ



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(Prob. 5.5)

The secondary maxima occur at θ for

which β=tanβ. β=1.43π, 2.46π,

3.47π, ...




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5.4 Fraunhofer diffraction for the rectangular aperture For a rectangular aperture of width a and height b, dA=dxdy.

220 )sin()sin(

ββ

ααII =

,sin21 ,sin

21 θβφα kbka ==

The minimum values occur for α=±π, ±2π, … and β=±π, ±2π, …



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5.4 Fraunhofer diffraction for the circular aperture For a circular aperture of radius R, dyyRdA 222 −=

220

21

0 )( where,)(2 RCIJII πρ

ρ=

=

0 as ,2/1/)(kind1st theoffunction Bessel)(

/)(1

sin ,

2

1

1

121

1

22sin0

→→=

=−

==

−=

∫

∫

+

−

+

−

ρρρρ

ρρπ

θρ

ρ

θ

JJ

Jduue

kRRyu

dyyReCeU

ui

R

R

ikyikr



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5.4 Fraunhofer diffraction for the circular aperture

aperture theofdiameter 2

22.1832.3sin

==

≈==

RD

DkRθλθ

The bright central area is known as the Airy disk.

1st zero of the Bessel function ρ=3.832.

The angular radius of the 1st dark ring is



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Optical Resolution

The image of a distant point source formed at the focal plane of a camera lens is a Fraunfoffer diffraction pattern for which the aperture is the lens opening D. Thus the image of a composite source is a superposition of many Airy disks. The resolution in the image depends on the size of the individual Airy disks. Rayleigh criterion: minimum angular separation between two equal point sources such that they can be just barely resolved. At this angular separation the central maximum of the image of one source falls on the 1st minimum of the other.

22.1D

λθ ≈

Rayleigh criterion for the resolution



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5.4 Fraunhofer diffraction for the double slit For a circular aperture of radius R, dyyRdA 222 −=

( )

( )

γβ

β

θγθβ

γβ

β

θ

θ

γβ

θθ

θθθ

θθθ

22

0

sinsin

sinsin)(sin

sin

0

sinsin

cossin

sin21 ,sin

21

cossin2

1sin

1

1sin1

=

==

=

+

−=

−+−=

+=

+

+

∫∫∫

II

khkb

ebe

eik

e

eeeik

dyedyedye

ii

ikhikb

ikbbhikikb

bh

h

ikyb iky

Aperture

iky



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5.4 Fraunhofer diffraction for the double slit The single-slit factor (sinβ/β)2 appears as the envelope for the interference firnges given by the term cos2γ. Bright fringes occur for γ=0, ±π, ±2π, … The angular separation between fringes is given by ∆γ=π.

2hkhλπθ =≈∆πθθγ =∆=∆ cos

21 kh



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5.4 Fraunhofer diffraction for the double slit




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5.4 Fraunhofer diffraction for the double slit




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Multiple slits, Diffraction gratings

[ ]

θγθβ

γγ

ββ

θ

θ

γβ

θ

θθ

θθθ

θθ

sin21 ,sin

21

sinsinsin

11

sin1

....1sin

1

....

)1(

sin

sinsin

sin)1(sinsin

)1(

)1(

sin2

20

sin

khkb

Nebe

ee

ike

eeik

e

dyedye

Nii

ikh

ikNhikb

hNikikhikb

bhN

hN

ikyhb

h

bh

h

b

Aperture

iky

==

=

−−

⋅−

=

+++−

=

++++=

−

−

+−

−

+

∫∫∫∫∫

22

0 sinsinsin

=

γγ

ββ

NNII

A diffraction pattern of a 633 nm laser through a grid of 150 slits https://en.wikipedia.org/wiki/Diffraction

The factor N has been inserted in order to normalize the expression, so that I=I0 when θ=0.



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The single-slit factor (sinβ/β)2 appears as the envelope of the diffraction pattern.

Principal maxima occur within the envelope for γ=nπ , n=0, π, 2π, …

θλ sinhn =

Secondary maxima occur for

γ=3π/2Ν, 5π/2Ν, …

Zeros occur for γ=π/Ν, 2π/Ν, …

n=order of diffraction

22

0 sinsinsin

=

γγ

ββ

NNII



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cos

, cos21

θγλθθθπγ

Nhkh

N=∆∴∆==∆

Resolving power of a grating spectroscope according to the Rayleigh criterion

NnRP =∆

=λ

λ

The angular width of a principal fringe is found by setting the change of Nγ equal to π.

θλ sinhn =

If N is made very large, then ∆θ is very small, and the diffraction pattern consists of a series of sharp fringes corresponding to the different orders n=0, ±π, ±2π, … For a given order the dependence of θ on the wavelength gives by differentiation

θλθ

coshn∆

=∆

For a typical grating with 600 lines/mm ruled over a total width of 10 cm, N=60,000 and the theoretical resolving power can be 60,000n.

chapter 5. diffraction part 1 - seoul national...

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