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Diffraction: Scalar wave theory

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Scalar theory of diffraction: Huygens construction

APPROXIMATION: We use theapproximation to neglect polarizations (scalar

theory) and consider the wave as a single

complex variable ! with angular frequency "

and wave-vector k0having a magnitude "/c in

the direction of the propagation.We will also omit the time dependent wave

factor.

Consider the amplitude observed in the point P arising from light emitted from a point Q and scattered by a plane mask R.

We assume that an element of area dS at the mask is disturbed by a wave !1 and that this point acts a secondary emitter of

strength

b fs!

1dS

Strength of re-emissionTransmission function (real or complex)

The coherence(phase) of the re-emission is important. It must be exactly related to the original disturbance !1 otherwisethe diffraction effect will change with time.

If we consider the wave emitted from the point source Q of strength aQ as a spherical wave

!1

=

aQ

d1exp j k

0

d1

( )

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Scalar theory of diffraction: Huygens construction

The area dS acts as a secondary emitter of

strength

b aS = b f

s!

1dS

Thus the contribution to ! received at P is

d!P = b f

s!

1d!1 exp j k

0d( )dS = b fsaQ d d1( )

!1

exp j k0

d+ d1( )( )dS

The total amplitude at P is therefore the integral over the whole plane of the screen R

!P = b a

Q f

s

R

!! d d1( )"1

exp j k0

d+ d1( )( )dS

Paraxial approximation b=1 in the forwarddirection and zero in the backward direction

The inclination factor b depends on the angle #

!

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Scalar theory of diffraction: paraxial approximation

We restrict our attention to a system illuminated by a plane wave. We take the source Q far away from the mask andincrease its intensity to keep the ratio

aQ

d1

=A constant

Now we assume that the diffracting mask coincides with a plane wavefront of the incident wave. The amplitude in theobserving screen becomes

!P = b a

Q f

s

R

!! d d1( )"1

exp j k0

d+ d1( )( )dS#!P = b A exp j k0z1( )

f r()dR

!! exp j k0 d( )d2r

Where we denote the position of S by r and thus fs is replaced by f(r)

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Because Q is very far way from the mask, the illumination is expected to be a plane wave and the constant phase factor

exp j k0

z1

( )can be absorbed into A. The intensity in the observation screen will be

I =!2

=!!*

!= b Af r( )

dR!! exp j k0d( )d

2r

Diffraction effects

Fresnel (near field). Phase term has non-linear terms larger than "/2

Fraunhofer (far field). The phase term varies linearly with r

Phase term that defines thediffraction regime

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To set quantitative limits for both regimes e can define a circle of radius $ which includes all the transmitting regions of themask. We now expand the phase term as

k0d = k

0 z

2+ r! p

2

( )1

2

! k0z+

1

2k0 z

!1r2! 2r . p + p

2( )+!

We see that this expression has

A constant term

A term linear in r

A quadratic term

k0 z+

1

2

p2

z

!

"#

$

%&

k0

zr . p

1

2

k0

r2

z

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The largest value of r that contributes to the problem is $. Thus the maximum size of the quadratic term is

1

2

k0

!2

z

The limit for Fresnel or Fraunhofer regimes is set by how big is the quadratic phase term compared to "

Fresnel or near field diffraction1

2k0

!2

z! "" !2 ! #z

Fraunhofer or far field diffraction1

2 k0

!2

z

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Fresnel diffraction

The integral to evaluate is

!= b Af r( )

dR!! exp j k0d( )d

2r

If we restrict to the case when the aperture is small compared with the distance z ($

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Fresnel diffraction

In order to evaluate the Huygens scattering strength factor we consider a very large aperture for which f(r) = 1 which willnot affect the propagation of the plane wave

!b A exp j k0z( )

2z

jk0

h R2( )exp j k0R2

2z

!

"#

$

%&' h 0( )

(

)**

+

,--'

dh

dsexp j k

0

s

2z

!

"#

$

%&ds

0

R2

./01

21

341

51

We can make the two following approximations:

1- R is large enough that h(R2) is negligible compared to h(0)

2- The integral is negligible because the integrand is a small function (proportional to z-3) multiplied by arapid oscillating term

Thus the expression above reduces to

! =!"b A exp j k0z( )

2z

jk0

h 0() =!Abexp j k0z( )2!

j k0

Since h 0( ) = f 0( )z =1

z

But we assumed that the aperture was very large, thus this result should be equal to the result of a free propagating planewave, namely

!=A exp j k0z( )

Comparing with equation (*)

(*)

b =! j k

0

2!

=!

j

"

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Fraunhofer diffraction

A plane wave traveling along the z direction and

incident at z = 0 in a mask with a transmission

f(x,y)

The diffracted light is collected by a lens with focal

distance f

Rays XB , OA , YC are parallel and focuses in P

Therefor the amplitude in P is the sum of the

amplitudes at X, O, etc with the appropriate phase

term

exp j k0XBP( ), exp j k0OAP( ),!

The amplitude in a point x,y in the screen (z = 0) is the product of the incident wave (assumed unity) times f(x,y)

To calculate the optical paths we use the Fermats principle: The optical paths from anypoint in a wavefront to its focus are all equal

If we represent the directions XB, OA, YC, by direction cosines then the wavefront normal to them through thepoint O which focuses in P is

XPP,OAP

,YCP

,!

!,m,n( )

! x + m y + n z= 0

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And the optical paths are all equal

The segment ZX is the projection of the distance OXonto the ray XB. It can be written as the component

of the vector (x,y,0) in the direction of

This is

OAP= ZBP=!

!,m,n( )

ZX = !x+m yThus

XBP=OAP! !x!m y

The amplitude in P is obtained integrating over the screen f x,y( ) exp j k0 XBP( )

!P =

exp j k0OAP( ) f x,y( ) exp !j k0 !

x+

m y( )( )"" dx dyAnd if we define u ! !k

0 v ! m k

0

! u,v( ) = exp j k0OAP( ) f x,y( ) exp !j u x+ v y( )( )"" dx dy

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! u,v( ) = exp j k0OAP( ) f x,y( ) exp !j u x+ v y( )( )"" dx dy

Conclusion:

The Fraunhofer diffraction pattern amplitude is given by the two-dimensional Fourier transform of the

mask transmission f(x,y)

The coordinates (u,v) can be related to the angles of diffraction through the vector!x

,!y

!,m,n( )! = sin!

xm = sin!

y Thus u = k

0sin!

x v = k

0sin!

y

The coordinates of a point in the screen

are known if we know the focal distance F

P= px,p

y( )

px =F

!

n! u

F

k0

py =F

m

n

!vF

k0

"

#

$$

%

$$

Paraxial

approximation

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Fraunhofer diffraction by a slit

f x,y( ) = rect x

a

!

"

#$

%

&=1 x '

a

2

0 x > a2

(

)

**

+**

a

! u

,v

( )=

exp !

j u x( )!a 2

a

2

! dx

exp !

j v y( )!"

"

# dy =

2 sin au2( )

u

! v

( )= a sinc au

2( )! v

( )

The intensity is obtained from the amplitude

! u,v( )2

= a2sinc

2 au

2( )

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Fraunhofer diffraction by a rectangular hole

Now consider a rectangular hole of sides a and b parallel to the x and y axis respectively

Since the function is the product of independent functions in x and y

And integrating

f x,y

( )= rect

x

a

!

"#

$

%&rect

y

b

!

"#

$

%&

! u,v( ) = exp j u x( )dx!

a

2

a

2

" exp j v y( )dy!b

2

b

2

"

! u,v( ) = ab sinc 1

2ua

!

"#

$

%&sinc

1

2vb

!

"#

$

%&

Documents

ece506 class 2014 41 to 56