Skip to content Skip to navigation Skip to collection information

OpenStax-CNX

You are here: Home » Content » Waves and Optics » Snell's Law

Navigation

Table of Contents

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? tag icon

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

This content is ...

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 display tagshide tags

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

    Comments:

    "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 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.
 

Snell's Law

Module by: Paul Padley. E-mail the author

Summary: We derive Snell's Law, and the Law of Reflection from the fundamental properties of an electromagnetic wave impinging upon an interface.

Snell's Law

Consider an electromagnetic wave impinging upon an interface: u ̂ n × E 0 i e i ( k i r + ω i t ) + u ̂ n × E 0 r e i ( k r r + ω r t + δ r ) = u ̂ n × E 0 t e i ( k t r + ω t t + δ t ) u ̂ n × E 0 i e i ( k i r + ω i t ) + u ̂ n × E 0 r e i ( k r r + ω r t + δ r ) = u ̂ n × E 0 t e i ( k t r + ω t t + δ t ) Where ( k i , ω i ) ( k i , ω i ) describes the incoming wave, ( k r , ω r , δ r ) ( k r , ω r , δ r ) the reflected wave, and ( k t , ω t , δ t ) ( k t , ω t , δ t ) the transmitted wave. At the interface (ie. at points where the vector r r points to the plane of the interface), all the waves must be in phase with each other. This means that the frequencies must all be equal and there can be no arbitrary phase between the waves. The net result of this is that we must have (for an interface passing through the origin): k i r = k r r = k t r k i r = k r r = k t r from which we get k i sin θ i = k r sin θ r . k i sin θ i = k r sin θ r .

It is important to note now that we are doing this at the interface. We have chosen a coordinate system so that the interface is at y = 0 y = 0 and contains the origin. This implies that the vector r r is lying in the plane of the interface at the point where we say that the above is true.

Finally, since the incident and reflected waves are in the same medium we must have k i = k r k i = k r and thus θ i = θ r θ i = θ r Also, we get that k i , k r , u ̂ k i , k r , u ̂ all line in a plane (because ( k i k r ) r = 0 ( k i k r ) r = 0 defines a plane). We also have u ̂ × ( k i k t ) = 0 u ̂ × ( k i k t ) = 0 and following the same arguments find that k i , k r , k t , u ̂ k i , k r , k t , u ̂ all line in a plane and that k i sin θ i = k t sin θ t . k i sin θ i = k t sin θ t . Now we know that ω i = ω t ω i = ω t so we can multiply both sides by c / ω i c / ω i and get n i sin θ i = n t sin θ t n i sin θ i = n t sin θ t

Digression, further justifying the above

At the interface, which we will set to y = 0 y = 0 for convenience (you can always switch back to any coordinate system afterwards. It is good practice to choose the coordinate system that makes your problem easy) [ k i r + ω i t ] | y = 0 = [ k r r + ω r t + δ r ] | y = 0 = [ k t r + ω t t + δ t ] | y = 0 [ k i r + ω i t ] | y = 0 = [ k r r + ω r t + δ r ] | y = 0 = [ k t r + ω t t + δ t ] | y = 0 now this must be true for all r r on the surface and for all t t so we must have ω i = ω r = ω t ω i = ω r = ω t So now we have [ k i r + ω i t ] | y = 0 = [ k r r + ω i t + δ r ] | y = 0 = [ k t r + ω i t + δ t ] | y = 0 [ k i r + ω i t ] | y = 0 = [ k r r + ω i t + δ r ] | y = 0 = [ k t r + ω i t + δ t ] | y = 0

which can be written

[ k i r ] | y = 0 = [ k r r + δ r ] | y = 0 = [ k t r + δ t ] | y = 0 [ k i r ] | y = 0 = [ k r r + δ r ] | y = 0 = [ k t r + δ t ] | y = 0

So now we can write [ ( k i k r ) r ] | y = 0 = δ r [ ( k i k r ) r ] | y = 0 = δ r

Since the interface passes through the origin, one of the allowed values of r r is 0. So this is only true if δ r = 0 δ r = 0 . (If the interface does not include the origin then you can not make this simplification, but clearly we can always choose a coordinate system such that this is true and thereby simplify our lives) likewise we could have written [ ( k i k r ) t ] | y = 0 = δ t [ ( k i k r ) t ] | y = 0 = δ t and applied the same argument to get δ t = 0 δ t = 0 . Lets just use this henceforth and thus write: [ k i r ] | y = 0 = [ k r r ] | y = 0 = [ k t r ] | y = 0 [ k i r ] | y = 0 = [ k r r ] | y = 0 = [ k t r ] | y = 0

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

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

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | More downloads ...

Add:

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? tag icon

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? tag icon

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