Download 6.2 Refraction

6.2 Refraction
• origin of refraction and the derivation of Snell's
law
• dispersion, wavelength-dependent refraction
and prisms
• how lenses use refraction to focus light
• the thin lens equation and image formation
• light gathering power and the lens f-number
• lens aberrations and how they affect the focal
spot size and image fidelity
6.2 : 1/13
Refraction
Refraction is the change in ___________________ which occurs
at a refractive index boundary. Two facts are responsible for
refraction: (1) the wavelength of light depends upon the
________________, λ = λ0/n; and, (2) at the refractive index
boundary the ________ of the wave has to be the same in both
materials.
The figure shows refraction for
an air/glass boundary. The
relationship between the two
angles can be determined by
geometry. The triangle ABC has
sinθ1 = BC/AC = (λ/n1)/AC. The
triangle CDA has sinθ2 = AD/AC
= (λ/n2)/AC. AC and λ can be
eliminated to obtain Snell's Law.
n1 sinθ1 = n2 sinθ2
6.2 : 2/13
n2 = 1.5
θ2
θ1
C
B
A
n1 = 1
D
Refraction Example (1)
When incident light is parallel to the surface normal, the
transmitted light is ____________ to the surface normal (i.e.
there is no refraction).
n1 × sin 0° = n2 × sin θ 2
sin θ 2 =
θ2 = 0
6.2 : 3/13
n1
×0 = 0
n2
n1 = 1
n2 = 1.5
Refraction Example (2)
When incident light is perpendicular to the surface normal, the
transmitted light travels at what is called the ______________.
n2 = 1.5
n1 × sin 90° = n2 × sin θc
sin θ c =
θc = sin
n1
×1
n2
θc = 41.8E
−1 ⎛ n1
⎞
⎜ ⎟
⎝ n2 ⎠
n1 = 1
6.2 : 4/13
θ = 90E
Refraction Example (3)
When propagating from a higher refractive index region into a
region with a lower refractive index, the largest angle that will be
transmitted is the critical angle. Light impinging on the
refractive index boundary at angles greater than the critical
angle will undergo ____________________. The wave in the low
index region is a __________________________ that extends
for a few wavelengths (this is analytically useful).
θ i = 65E
amplitude
n1 = 1
-300
-150
0
distance (nm)
6.2 : 5/13
n2 = 1.5
n2 = 1.5
150
300
n1 = 1
θr = 65E
Prism
• for a material with normal dispersion, violet light will get bent
toward the surface normal to a _________ extent than red light
• when the high index material is shaped into prism, light gets
refracted twice - with violet bending more than red both times
• different wavelengths of light will then exit the prism at
_____________
• wavelengths are dispersed by angle, which becomes ________
at a screen removed from the prism
air
n=1
white
red
glass
n = f(λ)
6.2 : 6/13
violet
Elliptical Lens
• when a plane wave impinges on an elliptically-shaped region of
higher refractive index, it is converted into a ________________
collapsing toward the focal point
• a lens can be formed by making the second surface spherical
• the distance from the front lens surface to the focal point is
called the _________________
• the focal spot size is
given by the following
elliptical surface
equation,
of lens
0.61λ f
ρ=
r
where ρ is the spot radius,
λ is the wavelength, f is
the focal length, and r is
the incoming beam radius
• ρ is called the
_____________________
6.2 : 7/13
ray tracing
focal point
spherical surface
of lens
plane wave
spherical wave
Spherical Lens
• elliptical surfaces are very difficult to grind
• close to the optical axis a spherical surface nicely approximates
an elliptical surface
• the approximation works only when sinθ = θ (this is called the
_________________________________________________)
region where
the two surfaces are nearly the
same
optical axis
elliptical surface
spherical surface
6.2 : 8/13
Thin Lenses
A thin lens is one where the physical thickness
is small compared to the surface radii of
curvature, the focal length, and the object and
image distances. Convex lenses cause
collimated light ____________, while concave
lenses cause collimated light _____________.
biconvex
plano-convex
biconcave
plano-concave
The focal length of a lens is determined by the
lens makers formula.
⎛ 1
1
1 ⎞
= ( nl − 1) ⎜
−
⎟
f
R
R
out ⎠
⎝ in
If a surface curves inward toward the right, its radius is ______.
If a surface curves inward toward the left, its radius is _______.
The focal length of a convex lens is then positive, while that for a
concave lens is negative.
6.2 : 9/13
Convex Lens Distances
f
d ob
dim
f
The relationship among the distances is given by,
where f is the focal length, dob is the distance of the object from
the lens, and dim is the distance of the image from the lens.
Note that the image is _______ and magnified by M = ________.
6.2 : 10/13
Concave Lens Distances
f
dob
f
dim
A concave lens creates a ____________. By convention f, dob
and dim are all ______________ in sign.
6.2 : 11/13
Lens f-Number
• the f-number of a lens is given by the focal length divided by
the diameter, f/# = f/D
• the f-number is used as a metric for the _____________ that
can be gathered from a point source - the lower the value the
higher the collection efficiency
• referring to the figure it can be seen that the fraction of light
collected is the area subtended by the lens divided by the area of
the sphere
f/#
1
2
5
10
20
% gathered
_____
1.5
0.25
0.062
0.016
light gathering power drops
approximately as the _______ of the
f-number
6.2 : 12/13
D
f
Lens Aberrations
• chromatic: Since the refractive index varies with wavelength,
the focal length will __________________. When an object is
multicolor the image will focus to a variety of distances. Chromatic
aberrations are minimized by combining convex and concave lenses into an
achromatic doublet.
• spherical: A spherical surface does not focus light to a single
point. The further a ray is from the optical axis, the closer it will
focus ____________. For approximately equal object and image distances,
the use of a lens with equally curved surfaces will minimize the aberration. For an
object or image at infinity, the use of a plano-convex lens will minimize the aberration
(the planar side is toward the focal spot).
• off-axis:
When the object is off-axis such that the ray bundle does
not satisfy sinθ = θ, the rays impinging on the outer edge of the
lens are focused ___________ from the optical axis than center
rays. This is called coma and gives comet-shaped images.
With an off-axis object the rays in the plane of the image
have a different focal distance than those perpendicular. This is
called astigmatism.
6.2 : 13/13