Skip to content Skip to navigation Skip to collection information

OpenStax_CNX

You are here: Home » Content » Waves and Optics » Phase Changes

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.
 

Phase Changes

Module by: Paul Padley. E-mail the author

Summary: We look at what happens to the phase of a wave upon reflection.

r ( E 0 r E 0 i ) = n i cos θ i n t cos θ t n i cos θ i + n t cos θ t r ( E 0 r E 0 i ) = n i cos θ i n t cos θ t n i cos θ i + n t cos θ t t ( E 0 t E 0 i ) = 2 n i cos θ i n i cos θ i + n t cos θ t t ( E 0 t E 0 i ) = 2 n i cos θ i n i cos θ i + n t cos θ t

r ( E 0 r E 0 i ) = n t cos θ i n i cos θ t n t cos θ i + n i cos θ t r ( E 0 r E 0 i ) = n t cos θ i n i cos θ t n t cos θ i + n i cos θ t t ( E 0 t E 0 i ) = 2 n i cos θ i n t cos θ i + n i cos θ t t ( E 0 t E 0 i ) = 2 n i cos θ i n t cos θ i + n i cos θ t We can rewrite these equations using Snell's Law to eliminate the cos θ t cos θ t term. From simple trigonometry we know that cos θ t = 1 sin 2 θ t . cos θ t = 1 sin 2 θ t . We also know from Snell's law that sin θ t = n i n t sin θ i sin θ t = n i n t sin θ i so we have cos θ t = 1 n i 2 n t 2 sin 2 θ i . cos θ t = 1 n i 2 n t 2 sin 2 θ i .

We can substitute this into r = n i cos θ i n t 1 n i 2 n t 2 sin 2 θ i n i cos θ i + n t 1 n i 2 n t 2 sin 2 θ i = n i cos θ i n t 2 n i 2 sin 2 θ i n i cos θ i + n t 2 n i 2 sin 2 θ i = cos θ i n t 2 n i 2 sin 2 θ i cos θ i + n t 2 n i 2 sin 2 θ i r = n i cos θ i n t 1 n i 2 n t 2 sin 2 θ i n i cos θ i + n t 1 n i 2 n t 2 sin 2 θ i = n i cos θ i n t 2 n i 2 sin 2 θ i n i cos θ i + n t 2 n i 2 sin 2 θ i = cos θ i n t 2 n i 2 sin 2 θ i cos θ i + n t 2 n i 2 sin 2 θ i

Similarly we can derive that

r = n t 2 n i 2 cos θ i n t 2 n i 2 sin 2 θ i n t 2 n i 2 cos θ i + n t 2 n i 2 sin 2 θ i r = n t 2 n i 2 cos θ i n t 2 n i 2 sin 2 θ i n t 2 n i 2 cos θ i + n t 2 n i 2 sin 2 θ i

t = 2 cos θ i cos θ i + n t 2 n i 2 sin 2 θ i t = 2 cos θ i cos θ i + n t 2 n i 2 sin 2 θ i

t = 2 n t n i cos θ i n t 2 n i 2 cos θ i + n t 2 n i 2 sin 2 θ i t = 2 n t n i cos θ i n t 2 n i 2 cos θ i + n t 2 n i 2 sin 2 θ i

This form allows us to easily plot the coefficients for different values of θ i . θ i . For example,here are the coefficients for the case where n t n i = 1.5 . n t n i = 1.5 .

Figure 1: The transmmission and reflection coefficients for the case where the ratio of transmitted to incident indices of refraction is 1.5. The top two curves are transmission. The lower two are reflection, with red being for the E field transverse to the plane of incidence.
Figure 1 (fresnel15.gif)
It is interesting to note that the sign of the coefficient can change on reflection. E r = | r | E = e i π | r | E 0 e i ( K r ω t ) = | r | E 0 e i ( K r ω t + π ) E r = | r | E = e i π | r | E 0 e i ( K r ω t ) = | r | E 0 e i ( K r ω t + π )

This corresponds to a phase change by π π upon reflection.

We can also look at what happens to the reflection coefficients when n i n t = 1.5 . n i n t = 1.5 .

Figure 2: The reflection coefficients for the case where the ratio of the incident to the transmitted incidence of reflection is 1.5.
Figure 2 (Fresnel2.gif)
That is going from a high index of refraction material to a lesser. In this case we see at the critical angle we get total internal reflection. What happens to the phase here is complicated.

When we have n i n t > 1 n i n t > 1 (or n t n i < 1 ) n t n i < 1 ) it is convenient to write r = cos θ i i sin 2 θ i n t 2 n i 2 cos θ i + i sin 2 θ i n t 2 n i 2 r = cos θ i i sin 2 θ i n t 2 n i 2 cos θ i + i sin 2 θ i n t 2 n i 2

Now to understand what this implies we need to digress a little. Recall that e i α = cos α + i sin α . e i α = cos α + i sin α . We could have written this as e i α = a + i b e i α = a + i b then we see that α = tan 1 b a . α = tan 1 b a . Now consider cos α i sin α cos α + i sin α = e i α e i α = e 2 i α cos α i sin α cos α + i sin α = e i α e i α = e 2 i α

Now looking back at r r we see that r = e i φ r = e i φ where tan φ 2 = sin 2 θ i n t 2 n i 2 cos θ i . tan φ 2 = sin 2 θ i n t 2 n i 2 cos θ i .

We could go through a similar excersize for r r and get the same result with

tan φ 2 = sin 2 θ i n t 2 n i 2 n t 2 n i 2 cos θ i . tan φ 2 = sin 2 θ i n t 2 n i 2 n t 2 n i 2 cos θ i . Figure 20-8 in Pedrotti and Pedrotti summarizes all the possible phase changes.

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 | 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 ...

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