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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

- Consider two well behaved functions U1’, U2’ that are solutions of the wave equation.
- Let U1’ = U1e-it ; U2’ = U2e-it
- Thus U1 , U2 are the spatial part of the functions and since,
- we have,

Green’s theorem

- Consider the product U1 grad U2 = U1U2
- Using Gauss’ Theorem
- Where S = surface enclosing V
- Thus,

Green’s Theorem

- Now expand left hand side,
- Do the same for U2U1 and subtract from (2), gives Green’s theorem

Green’s theorem

- Now for functions satisfying the wave equation (1), i.e.
- Consequently,
- since the LHS of (3) = 0

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

- 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

- 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)

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

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

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.

Fresnel-Kirchoff Formula

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

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

- 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

- 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.

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 diffractionS

P

Hecht 10.2

Hecht 10.3

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 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 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 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 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 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 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 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 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 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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