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Diffraction. Monday Nov. 18, 2002. Diffraction theory (10.4 Hecht). We will first develop a formalism that will describe the propagation of a wave – that is develop a mathematical description of Huygen’s principle. Wavefront U. What is U here?. We know this. Diffraction theory.

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Diffraction l.jpg

Diffraction

Monday Nov. 18, 2002


Diffraction theory 10 4 hecht l.jpg
Diffraction theory (10.4 Hecht)

  • We will first develop a formalism that will describe the propagation of a wave – that is develop a mathematical description of Huygen’s principle

Wavefront U

What is U here?

We know this


Diffraction theory l.jpg
Diffraction theory

  • Consider two well behaved functions U1’, U2’ that are solutions of the wave equation.

  • Let U1’ = U1e-it ; U2’ = U2e-it

  • Thus U1 , U2 are the spatial part of the functions and since,

  • we have,


Green s theorem l.jpg
Green’s theorem

  • Consider the product U1 grad U2 = U1U2

  • Using Gauss’ Theorem

  • Where S = surface enclosing V

  • Thus,


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Green’s Theorem

  • Now expand left hand side,

  • Do the same for U2U1 and subtract from (2), gives Green’s theorem


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Green’s theorem

  • Now for functions satisfying the wave equation (1), i.e.

  • Consequently,

  • since the LHS of (3) = 0


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Green’s theorem applied to spherical wave propagation

  • Let the disturbance at t=0 be,

    where r is measured from point P in V and U1 = “Green’s function”

  • Since there is a singularity at the point P, draw a small sphere P, of radius , around P (with P at centre)

  • Then integrate over +P, and take limit as   0


Spherical wave propagation l.jpg
Spherical Wave propagation

P

Thus (4) can be written,


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Spherical Wave propagation

  • In (5), an element of area on P is defined in terms of solid angle

  • and we have used

  • Now consider first term on RHS of (5)


Kirchoff s integral theorem l.jpg
Kirchoff’s Integral Theorem

  • Now U2 = continuous function and thus the derivative is bounded (assume)

  • Its maximum value in V = C

  • Then since eik  1 as 0 we have,

  • The second term on the RHS of (5)


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Kirchoff’s Integral Theorem

Now as 0 U2(r)  UP (its value at P)

and,

Now designate the disturbance U as an electric field E


Kirchoff integral theorem l.jpg
Kirchoff integral theorem

This gives the value of disturbance at P in terms of values on surface  enclosing P.

It represents the basic equation of scalar diffraction theory


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Geometry of single slit

Have infinite screen with aperture A

Let the hemisphere (radius R) and screen with aperture comprise the surface () enclosing P.

P

S

r

r’

’

Radiation from source, S, arrives at aperture with amplitude

Since R 

E=0 on .

R

Also, E = 0 on side of screen facing V.


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Fresnel-Kirchoff Formula

  • Thus E=0 everywhere on surface except the portion that is the aperture. Thus from (6)


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Fresnel-Kirchoff Formula

  • Now assume r, r’ >>  ; then k/r >> 1/r2

  • Then the second term in (7) drops out and we are left with,

Fresnel Kirchoff diffraction formula


Obliquity factor l.jpg
Obliquity factor

  • Since we usually have ’ = - or n.r’=-1, the obliquity factor F() = ½ [1+cos ]

  • Also in most applications we will also assume that cos   1 ; and F() = 1

  • For now however, keep F()


Huygen s principle l.jpg
Huygen’s principle

  • Amplitude at aperture due to source S is,

  • Now suppose each element of area dA gives rise to a spherical wavelet with amplitude dE = EAdA

  • Then at P,

  • Then equation (6) says that the total disturbance at P is just proportional to the sum of all the wavelets weighted by the obliquity factor F()

  • This is just a mathematical statement of Huygen’s principle.


Fraunhofer vs fresnel diffraction l.jpg

In Fraunhofer diffraction, both incident and diffracted waves may be considered to be plane (i.e. both S and P are a large distance away)

If either S or P are close enough that wavefront curvature is not negligible, then we have Fresnel diffraction

Fraunhofer vs. Fresnel diffraction

S

P

Hecht 10.2

Hecht 10.3


Fraunhofer vs fresnel diffraction19 l.jpg
Fraunhofer vs. Fresnel Diffraction waves may be considered to be plane (i.e. both S and P are a large distance away)

’

r’

r

h’

h

d’

S

P

d


Fraunhofer vs fresnel diffraction20 l.jpg
Fraunhofer Vs. Fresnel Diffraction waves may be considered to be plane (i.e. both S and P are a large distance away)

Now calculate variation in (r+r’) in going from one side of aperture to the other. Call it 


Fraunhofer diffraction limit l.jpg
Fraunhofer diffraction limit waves may be considered to be plane (i.e. both S and P are a large distance away)

sin’

sin

  • Now, first term = path difference for plane waves

’

sin’≈ h’/d’

sin ≈ h/d

sin’ + sin =  ( h’/d + h/d )

Second term = measure of curvature of wavefront

Fraunhofer Diffraction 


Fraunhofer diffraction limit22 l.jpg
Fraunhofer diffraction limit waves may be considered to be plane (i.e. both S and P are a large distance away)

  • If aperture is a square -  X 

  • The same relation holds in azimuthal plane and 2 ~ measure of the area of the aperture

  • Then we have the Fraunhofer diffraction if,

Fraunhofer or far field limit


Fraunhofer fresnel limits l.jpg
Fraunhofer, Fresnel limits waves may be considered to be plane (i.e. both S and P are a large distance away)

  • The near field, or Fresnel, limit is

  • See 10.1.2 of text


Fraunhofer diffraction l.jpg
Fraunhofer diffraction waves may be considered to be plane (i.e. both S and P are a large distance away)

  • Typical arrangement (or use laser as a source of plane waves)

  • Plane waves in, plane waves out

screen

S

f1

f2


Fraunhofer diffraction25 l.jpg
Fraunhofer diffraction waves may be considered to be plane (i.e. both S and P are a large distance away)

  • Obliquity factor

    Assume S on axis, so

    Assume  small ( < 30o), so

  • Assume uniform illumination over aperture

    r’ >>  so is constant over the aperture

  • Dimensions of aperture << r

    r will not vary much in denominator for calculation of amplitude at any point P

    consider r = constant in denominator


Fraunhofer diffraction26 l.jpg
Fraunhofer diffraction waves may be considered to be plane (i.e. both S and P are a large distance away)

  • Then the magnitude of the electric field at P is,


Single slit fraunhofer diffraction l.jpg
Single slit Fraunhofer diffraction waves may be considered to be plane (i.e. both S and P are a large distance away)

P

y = b

r

dy

ro

y

r = ro - ysin

dA = L dy

where L   ( very long slit)


Single slit fraunhofer diffraction28 l.jpg
Single slit Fraunhofer diffraction waves may be considered to be plane (i.e. both S and P are a large distance away)

Fraunhofer single slit diffraction pattern


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