Chapter 5. Diffraction Part 1

EE 430.423.001 2016. 2^nd Semester

2016. 10. 18.

Changhee Lee

School of Electrical and Computer Engineering Seoul National Univ.

chlee7@snu.ac.kr

Chapter 5. Diffraction

Part 1

EE 430.423.001 2016. 2^nd Semester

• 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

EE 430.423.001 2016. 2^nd Semester

Green’s theorem

) (

) (

) (

, F

, F F

theorem Divergence

) (

) (

2 2 2

V U

V U V

U U

V V U

dV dA

dV U V

U V

dA V U U V

n n

n

∇

⋅

∇ +

∇

=

∇

⋅

∇

∇

−

∇

=

⋅

∇

=

∇

∇

−

∇

=

∇

−

∇

∫∫∫

∫∫

∫∫∫

∫∫





 

5.2 Fundamental theory

2 2 2 2

2 2 2 2

1 1

t V V u

t U U u

∂

= ∂

∇

∂

= ∂

∇

0 )

( ∇ − ∇ =

∫∫

^{V nU U nV dA}

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

EE 430.423.001 2016. 2^nd Semester

5.2 Fundamental theory

r V e

V

t kr

i( )

0

ω

= +

Suppose that we take V to be the wave function

0 )

( )

( ² Ω =

∂

− ∂

∂

− ∂

∇

−

∇

∫∫

∫∫

^e_r^{ikr nU U n e}_r^{ikr dA e}_r^{ikr U}_r ^U _r ^e_r^{ikr r=ρρ d}

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.

P

Pd U

U Ω = 4π

∫∫

dA r U

e r

U e

U _n

ikr ikr

n

P ^{= −} ₄¹_π

∫∫

^{( ∇ − ∇ )}

Kirchhoff integral theorem

U = optical disturbance

EE 430.423.001 2016. 2^nd Semester

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 r

U e U

t kr i −ω

= The wave function U at the aperture



 



 −

∂ =

= ∂

∇



 



 −

∂ =

= ∂

∇

2 ' '

' '

2

' ) '

' , ' cos(

) ' ' , ' cos(

) , cos(

) , cos(

r e r

r ike r n

e r r

r n e

r e r

r ike r n

e r r

r n e

ikr ikr

ikr ikr

n

ikr ikr

ikr ikr

n

















r dA e r

e r

e r

e e

U U

ikr n ikr ikr

n ikr t

i

P ^{= 0}_4π^−ω

∫∫

^{( ∇} _'^{' −} _' ^{' ∇ )}

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

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

EE 430.423.001 2016. 2^nd Semester

Fresnel-Kirchhoff formula

[

^{n r n r}

]

^dA

rr e e

U ikU

r r ik t

i

P ^{= −} ₄⁰_π^−ω

∫∫

^{( +}_' ^{') cos(, )−cos(, ')}

[ ]

'

, 1 ) , 4 cos(

' 0

) (

r e U U

dA r

r n e U U ik

ikr A

t i kr i A P

=

+

−

= _π

∫∫

^{−ω  }

Fresnel-Kirchhoff integral formula

Circular aperture, r_'= _{constant, cos(}n_,r_') = −₁

=obliquity factor )]

' , cos(

) ,

[cos(n r n r

−

EE 430.423.001 2016. 2^nd Semester

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, U_P= U_1P + U_2P (Babinet’s principle)

If U_P=0, U_1P = - U_2P

The complementary apertures yield identical optical disturbances, except that they differ in phase by 180^o. The intensity at P is

therefore the same for the two apertures.

EE 430.423.001 2016. 2^nd Semester

Babinet’s principle

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

EE 430.423.001 2016. 2^nd Semester

http://userdisk.webry.biglobe.ne.jp/006/095/15/N000/000/004/136844073251013202889_Corona.JPG

Babinet’s principle

EE 430.423.001 2016. 2^nd Semester

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.

EE 430.423.001 2016. 2^nd Semester

5.3 Fraunhofer and Fresnel Diffraction

...

1) ' ( 1 2 ) 1

' ( '

' '

) (

) ' ( '

2

2 2

2 2

2 2

2 2

+ +

+ +

=

+

− +

− +

+ +

+ +

=

∆

δ δ

δ δ

d d

d h d

h

h d

h d

h d

h d

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

λ δ <<

+ 1) ² '

( 1 2 1

d d

Criterion for Fraunhofer diffraction

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

EE 430.423.001 2016. 2^nd Semester

5.4 Fraunhofer Diffraction Patterns

[

^{n r n r}

]

^dA

rr e e

U ikU

r r ik t

i

P ^{= −} ₄⁰_π^−ω

∫∫

^{( +}_' ^{') cos(, )−cos(, ')}

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) e^ikr’/r’ is nearly constant and can be taken outside the integral.

(3) The variation of e^ikr/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.

dA e

C

U_P ⁼

∫∫

^ikr

EE 430.423.001 2016. 2^nd Semester

5.4 Fraunhofer diffraction patterns for the single slit

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

EE 430.423.001 2016. 2^nd Semester

5.4 Fraunhofer diffraction patterns for the single slit

0 for

of value the

sin

0

0

=

=

+

=

y r

r

y r

r θ

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

CbL e

C kb

C

k kb L

Ce

Ldy e

Ce U

ikr ikr

b b ikr iky

0 0

0

' , 2 sin

1 sin ) ( '

sin

) 2 sin

sin(1 2

2 2

sin

=

=

=

=

=

∫

−⁺

θ β

β

β θ

θ

θ

EE 430.423.001 2016. 2^nd Semester

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 1^st minimum, β=π, sinθ=2π/kb=λ/b.

slit the

of area 0,

for irradiance

sin ) (

2 0

2 0

2

∝

=

=

=

=

CbL I

I U

I

θ β

β

EE 430.423.001 2016. 2^nd Semester

5.4 Fraunhofer diffraction patterns for the single slit

(Prob. 5.5)

The secondary maxima occur at θ for which β=tanβ. β=1.43π, 2.46π,

3.47π, ...

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

EE 430.423.001 2016. 2^nd Semester

5.4 Fraunhofer diffraction for the rectangular aperture

For a rectangular aperture of width a and height b, dA=dxdy.

2 2

0 sin )

( sin )

( β

β α

I α I =

, 2 sin

1 , 2 sin

1 φ β θ

α = ka = kb

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

EE 430.423.001 2016. 2^nd Semester

5.4 Fraunhofer diffraction for the circular aperture

For a circular aperture of radius R, _dA = _{2 R}² − _y²_dy

2 2 0

2 1

0 2 ( ) , where ( )

R C J I

I

I π

ρ

ρ  =



 



= 

0 as

, 2 / 1 /

) (

kind 1st

the of function Bessel

) (

/ ) ( 1

sin

,

2

1 1

1 1 2

1

2 2

0 sin

→

→

=

=

−

=

=

−

=

∫

∫

+

−

+

−

ρ ρ

ρ ρ

ρ ρ π

θ ρ

ρ

θ

J J

J du

u e

R kR u y

dy y R

e Ce

U

u i

R R

iky ikr

EE 430.423.001 2016. 2^nd Semester

5.4 Fraunhofer diffraction for the circular aperture

aperture the

of diameter 2

22 . 1 832 . sin 3

=

=

≈

=

= R D

D

kR λ θ

θ

The bright central area is known as the Airy disk.

1^st zero of the Bessel function ρ=3.832.

The angular radius of the 1^st dark ring is

EE 430.423.001 2016. 2^nd Semester

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 1^st minimum of the other.

22 . 1

D θ ≈ λ

Rayleigh criterion for the resolution

EE 430.423.001 2016. 2^nd Semester

5.4 Fraunhofer diffraction for the double slit

For a circular aperture of radius R, _dA = _{2 R}² − _y²_dy

( )

( )

β γ β

θ γ

θ β

β γ β θ

θ

γ β

θ θ

θ θ

θ

θ θ

θ

2 2

0

sin sin

sin sin

) ( sin

sin 0

sin sin

sin cos

2 sin 1

, 2 sin

1

sin cos 2

sin 1 1 sin 1

1



 



= 

=

=



 



= 

 +



 



 −

=

− +

−

=

+

=

+

∫

+

∫

∫

I I

kh kb

e be

ik e e

e e

ik e

dy e

dy e

dy e

i i

ikh ikb

ikb b

h ik ikb

b h h b iky

iky Aperture

iky

EE 430.423.001 2016. 2^nd Semester

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.

2

h kh

λ θ ≈ π = π ∆

θ θ

γ = ∆ =

∆ cos

2 1 kh

EE 430.423.001 2016. 2^nd Semester

5.4 Fraunhofer diffraction for the double slit

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

EE 430.423.001 2016. 2^nd Semester

5.4 Fraunhofer diffraction for the double slit

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

EE 430.423.001 2016. 2^nd Semester

Multiple slits, Diffraction gratings

[ ]

θ γ

θ β

γ γ β

β θ

θ

γ β

θ θ θ

θ θ θ

θ θ

2 sin 1

, 2 sin

1

sin sin sin

1 1 sin

1

....

sin 1 1

....

) 1 (

sin sin sin

sin ) 1 ( sin

sin

) 1 (

) 1 ( 2 sin

2 0

sin

kh kb

e N be

e e ik

e

e ik e

e

dy e

dy e

N i i

ikh ikNh ikb

h N ik ikh

ikb

b h N

h N hb iky

h b

h h b

Aperture iky

=

=



 







 



= 

−

⋅ −

= −

+ +

− +

=

+ +

+ +

=

−

−

+

−

−

+

∫ ∫

∫

∫

∫

2 2

0 sin

sin

sin 



 



 



 



= 

γ γ β

β

N I N

I

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.

EE 430.423.001 2016. 2^nd Semester

Multiple slits, Diffraction gratings

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, …

θ λ hsin n =

Secondary maxima occur for γ=3π/2Ν, 5π/2Ν, …

Zeros occur for γ=π/Ν, 2π/Ν, …

n=order of diffraction

2 2

0 sin

sin

sin 



 



 



 



= 

γ γ β

β

N I N

I

EE 430.423.001 2016. 2^nd Semester

Multiple slits, Diffraction gratings

cos

, 2 cos

1

θ θ γλ

θ π θ

γ kh Nh

N = ∆ ∴∆ =

=

∆

Resolving power of a grating spectroscope according to the Rayleigh criterion

Nn

RP =

= ∆ λ λ

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

θ λ hsin n =

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 θ

θ λ

cos h

= n∆

∆

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.