Connexions

You are here: Home » Content » Waves and Optics » Diffraction from a Circular Aperture

• Oscillations in Mechanical Systems

• Partial Derivatives

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
• Rice Digital Scholarship

This collection is included in aLens by: Digital Scholarship at Rice University

"This book covers second year Physics at Rice University."

Click the "Rice Digital Scholarship" link to see all content affiliated with them.

Click the tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Inside Collection (Course):

Course by: Paul Padley. E-mail the author

Diffraction from a Circular Aperture

Module by: Paul Padley. E-mail the author

Summary: We examine diffraction through a circular aperture.

Circular Aperture

The circular aperture is particularly important because it is used a lot in optics. A telescope typically has a circular aperture for example.

We can use the same expression for the E field that we had for the rectangular aperture for any possible aperture, as long as the limits of integration are appropriate. So we can write

E = ɛ A R e i ( k R ω t ) a p e r t u r e e i K ( Y y + Z z ) / R y z E = ɛ A R e i ( k R ω t ) a p e r t u r e e i K ( Y y + Z z ) / R y z

For a circular aperture this integration is most easily done with cylindrical coordinates. Look at the figure

Then we have z = ρ cos φ z = ρ cos φ y = ρ sin φ y = ρ sin φ Z = q cos Φ Z = q cos Φ Y = q sin Φ Y = q sin Φ Then Y y + Z z = ρ q cos φ cos Φ + ρ q sin φ sin Φ Y y + Z z = ρ q cos φ cos Φ + ρ q sin φ sin Φ or Y y + Z z = ρ q cos ( φ Φ ) Y y + Z z = ρ q cos ( φ Φ ) and the integral becomes E = ɛ A R e i ( k R ω t ) 0 a 0 2 π e i K ρ q cos ( φ Φ ) / R ρ ρ φ E = ɛ A R e i ( k R ω t ) 0 a 0 2 π e i K ρ q cos ( φ Φ ) / R ρ ρ φ

In order to do this integral we need to learn a little about Bessel functions.

J 0 ( u ) = 1 2 π 0 2 π e i u cos v v J 0 ( u ) = 1 2 π 0 2 π e i u cos v v Is the definition of a Bessel function of the first kind order 0. J m ( u ) = 1 2 π 0 2 π e i ( m v + u cos v ) v J m ( u ) = 1 2 π 0 2 π e i ( m v + u cos v ) v Is the definition of a Bessel function of the first kind order m.

They have a number of interesting properties such as the recurrence relations u [ u m J m ( u ) ) ] = u m J m 1 ( u ) u [ u m J m ( u ) ) ] = u m J m 1 ( u ) so that for example when m = 1 m = 1 0 u u J 0 ( u ) u = u J 1 ( u ) . 0 u u J 0 ( u ) u = u J 1 ( u ) . In order to numerically calculate the value of a Bessel function one uses the expansion J m ( x ) = s = 0 ( 1 ) s s ! ( m + s ) ! ( x 2 ) m + 2 s . J m ( x ) = s = 0 ( 1 ) s s ! ( m + s ) ! ( x 2 ) m + 2 s .

Now we want to evaluate the integral E = ɛ A R e i ( k R ω t ) 0 a 0 2 π e i K ρ q cos ( φ Φ ) / R ρ ρ φ E = ɛ A R e i ( k R ω t ) 0 a 0 2 π e i K ρ q cos ( φ Φ ) / R ρ ρ φ which we can do at any value of Φ Φ since the problem is symmetric about Φ Φ . So we can simplify things greatly if we do the integral at Φ = 0 Φ = 0 E = ɛ A R e i ( k R ω t ) 0 a 0 2 π e i K ρ q cos ( φ ) / R ρ ρ φ E = ɛ A R e i ( k R ω t ) 0 a 0 2 π e i K ρ q cos ( φ ) / R ρ ρ φ which becomes E = ɛ A R e i ( k R ω t ) 2 π 0 a J 0 ( K ρ q / R ) ρ ρ E = ɛ A R e i ( k R ω t ) 2 π 0 a J 0 ( K ρ q / R ) ρ ρ

Now J 0 J 0 is an even function so we can drop the minus sign and rewrite the expression as E = ɛ A R e i ( k R ω t ) 2 π 0 a J 0 ( K ρ q / R ) ρ ρ E = ɛ A R e i ( k R ω t ) 2 π 0 a J 0 ( K ρ q / R ) ρ ρ

To do this integral we change variables w = k ρ q / R w = k ρ q / R ρ = w R k q ρ = w R k q d ρ = R k q w d ρ = R k q w so that 0 a J 0 ( K ρ q / R ) ρ ρ = 0 k a q / R ( R k q ) 2 J 0 ( w ) w w = ( R k q ) 2 ( k a q R ) J 1 ( k a q / R ) = a 2 ( R k a q ) J 1 ( k a q / R ) = a 2 J 1 ( k a q / R ) k a q / R 0 a J 0 ( K ρ q / R ) ρ ρ = 0 k a q / R ( R k q ) 2 J 0 ( w ) w w = ( R k q ) 2 ( k a q R ) J 1 ( k a q / R ) = a 2 ( R k a q ) J 1 ( k a q / R ) = a 2 J 1 ( k a q / R ) k a q / R

So finally we have the result E = ɛ A e i ( k R ω t ) R 2 π a 2 J 1 ( k a q / R ) k a q / R E = ɛ A e i ( k R ω t ) R 2 π a 2 J 1 ( k a q / R ) k a q / R Or recognizing that π a 2 π a 2 is the area of the aperture A A and squaring to get the intensity we write I = I 0 [ 2 J 1 ( k a q / R ) k a q / R ] 2 I = I 0 [ 2 J 1 ( k a q / R ) k a q / R ] 2 If you want to write this in terms of the angle θ θ then one uses the fact that q / R = sin θ q / R = sin θ I ( θ ) = I ( 0 ) [ 2 J 1 ( k a sin θ ) k a sin θ ] 2 I ( θ ) = I ( 0 ) [ 2 J 1 ( k a sin θ ) k a sin θ ] 2

Above is a plot of the function J 1 ( x ) / x J 1 ( x ) / x . Notice how it peaks at 1 / 2 1 / 2 which is why there is the factor of two in the expression for the irradiance. Below is a 3D plot of the same thing (ie. J 1 ( r ) / r J 1 ( r ) / r ). Notice the rings.

Above is a plot of ( J 1 ( r ) / r ) 2 ( J 1 ( r ) / r ) 2 which corresponds to the irradiance one sees. The central peak out to the first ring of zero is called the Airy disk. This occurs at J 1 ( r ) / r = 0 J 1 ( r ) / r = 0 which can be numerically evaluated to give r = 3.83 r = 3.83 for the first ring.

For our circular aperture above this means the first zero occurs at k a q 1 / R = 3.83 k a q 1 / R = 3.83 or 2 π λ a q 1 R = 3.83 2 π λ a q 1 R = 3.83 q 1 = 1.22 R λ 2 a q 1 = 1.22 R λ 2 a In our case a a is the radius of the aperture and we can rewrite the expression using the diameter D = 2 a D = 2 a q 1 = 1.22 λ R / D q 1 = 1.22 λ R / D

Light passing through any circular aperture is going to be diffracted in this manner and this sets the limit of resolution on an optical device such as a telescope. Say one is trying resolve two sources, we can say the limit of resolution is when the central spot of one Airy disk is on the zero of the other Airy disk. This is known as the Raleigh critereon. While it is possible to define other crtieria, this is the most commenly used. See for example the plots below

In the above plot, the two sources can clearly be resolved. In the plot below, the two sources are going to be difficult to resolve.

So we say that the limit of our resolution occurs when the distance Δ q Δ q between two sources is Δ q = 1.22 R λ / D Δ q = 1.22 R λ / D or in the small angle limit Δ θ = Δ q / R Δ θ = Δ q / R Δ θ = 1.22 λ / D Δ θ = 1.22 λ / D

Content actions

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks