$prism and refraction$

Image 1 Caption: A diagram illustrating the refraction of monochromatic light rays by a prism of made from 5 of the 6 conventional spectral color terms (orange is omitted). The black line segments are normals (i.e., the perpendiculars) to the glass medium interface.

Features:

Monochromatic light is an ideal limit that CANNOT be reached in practice. By monochromatic light, one means either an ideal limit of single wavelength light or polychromatic light with a very narrow wavelength band.
The 6 conventional spectral color terms are:
The light rays in Image 1 have already been dispersed, but if you imagine them as superimposed, they would be dispersed by the prism and the light rays would be heading in different directions. Far from the prism (i.e., in the far field limit), the light rays would be well separated into a continuous spectrum created by dispersion by the prism.
Prisms disperse light using refraction which is governed by Snell's law (AKA the law of refraction). In many scientific applications, greater and better dispersion is obtained using a diffraction grating (which relies on diffraction rather than refraction) rather than a prism. A compact disc is incidentally acts as a crude diffraction grating. That is NOT its function, but it disperses light rather prettily. By the by, nowadays the compact disc is a retro technology (Wikipedia: Compact disc: Current_status). For more on compact discs, see Optics file: diffraction_compact_disk.html.
Image 2 Caption: An animation of how a prism disperses a white light light ray into a continuous spectrum.
In the far field limit, the dispersed continuous spectrum will be well spread out.
Actually, the prism disperses light beyond the visible band (fiducial range 0.4--0.7 μm) (i.e., into the ultraviolet band (fiducial range 0.01--0.4 μm) and infrared band (fiducial range 0.7 μm -- 0.1 cm)), but the human eye does NOT notice that. For example, a prism made of fused quartz (AKA quartz glass, silica glass) disperses light beyond the visible band (fiducial range 0.4--0.7 μm). Note for fused quartz (AKA quartz glass, the best transmittance range is 0.18--2.7 μm (Wikipedia: Fused quartz: List of physical properties).
$refraction simple diagram$
Image 3 Caption: An illustration of Snell's law (AKA the law of refraction) with refractive index n_2 > n_1. The explication of Snell's law is given below.
Snell's law (AKA the law of refraction) is
```
  n_1*sin(θ_1) = n_2*sin(θ_2)  , 
```
where the n_i are the refractive indexes n_i = c/v_i (n_i ≥ 1 always), vacuum light speed c = 2.99792458*10**8 m/s (exact by definition) ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns, v_i is the light speed in optical medium i (v_i ≤ c always), and the θ_i are the angles (i.e., incidence angle and refraction angle) of the light ray from the normal (i.e., the perpendicular) to the medium interface.
Note
```
sin(θ_2) = (n_1/n_2)*sin(θ_1)  , 
```
and so θ_2 subceeds/exceeds θ_1 if n_2 is greater/lesser than n_1.
In particular, note that a light ray bends toward a normal going from medium 1 to 2 if n_2 is greater than n_1.
Most common transparent solids and liquids have n_i greater than air's n_air, and so light rays bend toward/away the normal when entering/leaving these common materials when they are embedded in air.
Note incidence angles and refraction angles greater than 90° do NOT happen definitionally and have NO meaning in Snell's law. But what happens for case of incidence angles greater than those that give refraction angles of 90°? Total internal reflection which is illustrated by Image 5 below and which we discuss below Image 5.
Refractive indexes are generally wavelength dependent. In particular, common transparent solids and liquids have refractive indexes that are noticeably wavelength dependent.
For prisms (which are made from some kind of optical glass), the refractive index decreases with increasing wavelength. Thus, in Image 1 and Image 2, violet light refracts more than the red light.
How does a prism's work?
Image 1 and Image 2 actually tell all. But we can supplement the images with some description:
1. A single polychromatic incident light ray (and therefore an incident light ray of mixed wavelengths/frequencies) on the prism's surface at a nonzero incidence angle undergoes dispersion into continuum colored light rays at a continuum of refraction angles inside the prism because of the wavelength dependence of the refractive index.
2. Since the optical glass has a higher refractive index than air, the refracted light rays are bent toward the normal.
3. Because of the angling of the prism's medium interfaces, the already dispersed colored light rays are further dispersed when they exit the prism. So there will be a dispersed spectrum.
4. Note on exiting, the refracted colored light rays are bent away from the normal, but the aforesaid angling now gives an increase in dispersion.
5. What if a continuum of polychromatic light rays (i.e., a light beam) were incident on the prism?
  Exiting the prism at short range there is a strong remixing the of the colored light rays, and so the overall emergent light beam will be polychromatic though with a complexly different mixture than the incident light beam.
  However, at long range from the prism, the dispersed colored light beams will spread out because they emerge at different angles. So again there will be dispersed spectrum.
What if the sides of an optical medium object on which there are incident light rays are parallel? For an answer, consider Image 4 and its explication below
By the by, an common example of such an optical medium object is plate glass. $light ray refracted through plastic plate$
Image 4 Caption: A monochromatic light ray (maybe produced by a yellow laser, but maybe just well-collimated yellow light ray) being refracted twice by a plastic slab. Overlay description:
1. First incidence angle and second refraction angle = α.
2. First refraction angle and second incidence angle = β.
3. Slab thickness = d.
As Image 4 shows, if the sides of the slab are parallel, the twice refracted monochromatic light ray is displaced, but its direction is NOT changed.
Note:
1. If a polychromatic light ray was used, the light ray would be dispersed into a set of parallel colored light rays propagating in the same direction as the incident light ray.
2. What if a continuum of polychromatic light rays (i.e., a light beam) were incident on the slab?
  Exiting the slab, the colored light rays would be remixed and the exiting light beam would be polychromatic.
3. In the case where an incident light beam was homogeneous and a slab sufficiently long, the exiting light beam would have the same polychromatic nature as the incident light beam. This case is actually what one usually gets with plate glass.
4. The lenses of eyeglasses do NOT have parallel sides. Thus, there should be some differential refraction for polychromatic light beams (which effect in the special case of focusing for lenses is called chromatic aberration). Eyeglass wearers will thus often see a bit of dispersion for light beams incident on their lenses are large angles from the normal. Typically, the dispersion occurs for looking at light sources near the edges of the lenses.
Image 5 Caption: An illustration showing Snell's law and total internal reflection (with its critical angle θ_critical). Total internal reflection is when a light beam in one optical medium undergoes a reflection at a medium interface with NO refraction into a second optical medium which is, in fact, intrinsically transparent. Total internal reflection happens when the incident incidence angle of a light ray exceeds the critical angle θ_critical for total internal reflection. Total internal reflection is often used in optical devices. For example, see Optics file: optics_prism_porro_double.html.
A derivation of total internal reflection is beyond our scope, but a derivation of the formula for the critical angle θ_critical is given below.
A light ray from a optical medium 1 CANNOT propagate into a optical medium 2 if the refraction angle is greater than 90° since then it is NOT going into optical medium 2 at all. However, the refraction angle can be as large a 90° and this is the largest it can be. The incidence angle that gives refraction angle 90° turns out to be the critical angle θ_critical for total internal reflection---which is something we CANNOT prove, but we can derive the formula for the critical angle θ_critical from Snell's law
```
  n_1*sin(θ_1) = n_2*sin(θ_2)  . 
```
Say θ_2 = 90°, then sin(θ_2) = 1, θ_critical = θ_1, and
```
  θ_critical = arcsin(n_2/n_1)  . 
```
Since arcsin(x) is undefined for x > 1, there is only a critical angle for the case of n_2/n_1 ≤ 1: i.e., for the optical medium 1 having having the higher refractive index than optical medium 2.
Total internal reflection occurs in optical medium 1 at the medium interface for θ_1 > θ_critical. Total internal reflection is, in fact, specular reflection (if the medium interface is sufficiently smooth) where reflection angle equals the incidence angle as illustrated in Image 5.

Images:

Credit/Permission: © User:RJHall, 2005 / Creative Commons CC BY-SA 2.0.
Image link: Wikimedia Commons: File:Refraction varies by frequency.gif.
Credit/Permission: Lucas V. Barbosa (AKA User:LucasVB), 2007 / Public domain.
Image link: Wikipedia: File:Light dispersion conceptual waves.gif.
Credit/Permission: User:Oleg Alexandrov, 2007 / Public domain.
Image link: Wikimedia Commons: File:Snells law2.svg.
Credit/Permission: © User:MikeRun, 2019 / CC BY-SA 4.0.
Image link: Wikimedia Commons: File:Refraction-photo-with-overlay.png.
Credit/Permission: © User:Jfmelero, User:Gavin R Putland, 2019 / CC BY-SA 3.0.
Image link: Wikimedia Commons: File:ReflexionTotal en.svg.

local link: refraction_prism.html

Optics file

refraction_prism.html