Geometric Optics
1.Refraction of Light
Refraction is the bending of a wave when it enters a medium where its
speed is different. The refraction of light when it passes from a fast medium
to a slow medium bends the light ray toward the normal to the boundary
between the two media. The amount of bending depends on the indices of
refraction of the two media and is described quantitatively by Snell's Law.
Refraction is responsible for
image formation by lenses and
the eye.
The bending of refraction can be
visualized in terms of Huygens'
principle. As the speed of light is
reduced in the slower medium, the
wavelength is shortened
proportionately. The frequency is
unchanged; it is a characteristic of
the source of the light and unaffected
by medium changes.
1.1 Index of Refraction
The index of refraction is defined as the speed of light in vacuum
divided by the speed of light in the medium.
The indices of refraction of some common substances are given below
with a more complete description of the indices for optical glasses given
elsewhere. The values given are approximate and do not account for the
small variation of index with light wavelength which is called dispersion.
Material
n
Material
n
Vacuum
1.000
Ethyl alcohol
1.362
Air
1.000277
Glycerine
1.473
Water
4/3
Ice
1.31
Carbon
disulfide
1.63
Polystyrene
1.59
Methylene
iodide
1.74
Crown glass
1.50-1.62
Diamond
2.417
Flint glass
1.57-1.75
1.2 Snell's Law
Snell's Law relates the indices of refraction n of the two media to the
directions of propagation in terms of the angles to the normal. Snell's
law can be derived from Fermat's Principle or from the Fresnel
Equations.
If the incident medium has the larger index of refraction, then the angle
with the normal is increased by refraction. The larger index medium is
commonly called the "internal" medium, since air with n=1 is usually the
surrounding or "external" medium. You can calculate the condition for
total internal reflection by setting the refracted angle = 90° and calculating
the incident angle. Since you can't refract the light by more than 90°, all
of it will reflect for angles of incidence greater than the angle which gives
refraction at 90°.
1.3 Total Internal Reflection
When light is incident upon a medium of lesser index of refraction, the
ray is bent away from the normal, so the exit angle is greater than the
incident angle. Such reflection is commonly called "internal reflection".
The exit angle will then approach 90° for some critical incident angle θc,
and for incident angles greater than the critical angle there will be total
internal reflection.
The critical angle can be calculated from Snell's law by setting the refraction
angle equal to 90°. Total internal reflection is important in fiber optics and is
employed in polarizing prisms.
1.4 Law of Reflection
A light ray incident upon a reflective surface will be reflected at an angle
equal to the incident angle. Both angles are typically measured with respect
to the normal to the surface. This law of reflection can be derived from
Fermat's principle.
The law of reflection gives the familiar reflected image in a plane
mirror where the image distance behind the mirror is the same as the
object distance in front of the mirror.
1.5 Fermat's Principle:Reflection
Fermat's Principle: Light follows the path of least time. Of course the
straight line from A to B is the shortest time, but suppose it has a single
reflection. The law of reflection can be derived from this principle as follows:
The pathlength from A to B is
Since the speed is constant, the minimum
time path is simply the minimum distance
path. This may be found by setting the
derivative of L with respect to x equal to
zero.
This reduces to
which is
1.6 Fermat's Principle and Refraction
Fermat's Principle: Light follows the path of least time. Snell's Law can
be derived from this by setting the derivative of the time =0. We make
use of the index of refraction, defined as n=c/v.
1.7 Fresnel's Equations
Fresnel's equations describe the reflection and transmission of
electromagnetic waves at an interface. That is, they give the reflection
and transmission coefficients for waves parallel and perpendicular to the
plane of incidence. For a dielectric medium where Snell's Law can be
used to relate the incident and transmitted angles, Fresnel's Equations
can be stated in terms of the angles of incidence and transmission.
Fresnel's equations give the
reflection coefficients:
and
The transmission coefficients
are
and
2 Principal Focal Length
For a thin double convex lens, refraction acts to focus all parallel rays to a
point referred to as the principal focal point. The distance from the lens to
that point is the principal focal length f of the lens. For a double concave lens
where the rays are diverged, the principal focal length is the distance at
which the back-projected rays would come together and it is given a
negative sign. The lens strength in diopters is defined as the inverse of the
focal length in meters. For a thick lens made from spherical surfaces, the
focal distance will differ for different rays, and this change is called spherical
aberration. The focal length for different wavelengths will also differ slightly,
and this is called chromatic aberration.
The principal focal length of a lens is determined by the index of
refraction of the glass, the radii of curvature of the surfaces, and the
medium in which the lens resides. It can be calculated from the lensmaker's formula for thin lenses.
This shows parallel beams from two helium-neon lasers converging to
the principal focal point of a 30 cm double convex lens. The rays then
enter a diverging lens of focal length -10cm on the right. The laser
beams were made visible with a spray can of artificial smoke.
2.1 Focal Length and Lens Strength
The most important characteristic of a lens is its principal focal
length, or its inverse which is called the lens strength or lens
"power". Optometrists usually prescribe corrective lenses in terms of
the lens power in diopters. The lens power is the inverse of the focal
length in meters: the physical unit for lens power is 1/meter which is
called diopter.
2.2 Spherical Aberration球差
For lenses made with spherical surfaces, rays which are parallel to the optic axis
but at different distances from the optic axis fail to converge to the same point. For
a single lens, spherical aberration can be minimized by bending the lens into its
best form. For multiple lenses, spherical aberrations can be canceled by
overcorrecting some elements. The use of symmetric doublets like the orthoscopic
doublet greatly reduces spherical aberration.
When the concept of principal focal length is used, the presumption is that all
parallel rays focus at the same distance, which is of course true only if there are
no aberrations. The use of the lens equation likewise presumes an ideal lens, and
that equation is practically true only for the rays close to the optic axis, the socalled paraxial rays. For a lens with spherical aberration, the best approximation to
use for the focal length is the distance at which the difference between the
paraxial and marginal rays is the smallest. It is not perfect, but the departure from
perfect focus forms what is called the "circle of least confusion". Spherical
aberration is one of the reasons why a smaller aperture (larger f-number) on a
camera lens will give a sharper image and greater depth of field since the
difference between the paraxial and marginal rays is less.
2.3 Chromatic Aberration 色差
A lens will not focus different colors in exactly the same place because the focal
length depends on refraction and the index of refraction for blue light (short
wavelengths) is larger than that of red light (long wavelengths). The amount of
chromatic aberration depends on the dispersion of the glass.
One way to minimize this aberration is to use glasses of different dispersion
in a doublet or other combination. Another approach uses spaced doublets.
Doublet for Chromatic Aberration
The use of a strong positive lens made from a low dispersion glass like
crown glass coupled with a weaker high dispersion glass like flint glass can
correct the chromatic aberration for two colors, e.g., red and blue.
Such doublets are often cemented together (called achromat doublets) and
may be used in compound lenses such as the orthoscopic doublet.
Achromat Doublets消色差双片
An achromat doublet does not completely eliminate chromatic aberration,
but can eliminate it for two colors, say red and blue. The idea is to use a
lens pair with the strongest lens of low dispersion coupled with a weaker
one of high dispersion calculated to match the focal lengths for two
chosen wavelengths. Cemented doublets of this type are a mainstay of
lens design.
If the powers of the lenses of the doublet give a focus point in front of
the doublet as shown above, it is said to be a positive achromat.
Chromatic aberration for three colors can be eliminated with and
apochromat triplet.
Real Image Formation
If a luminous object is placed at a distance greater than the focal length away
from a convex lens, then it will form an inverted real image on the opposite side
of the lens. The image position may be found from the lens equation or by
using a ray diagram provided that it can be considered a "thin lens".
If the lens equation yields a negative image distance, then the image is a
virtual image on the same side of the lens as the object. If it yields a
negative focal length, then the lens is a diverging lens rather than the
converging lens in the illustration. The lens equation can be used to
calculate the image distance for either real or virtual images and for
either positive on negative lenses. The linear magnification relationship
allows you to predict the size of the image.
Virtual Image Formation
Diverging lenses form reduced, erect, virtual images. Using the
common form of the lens equation, f, P and i are negative quantities.
If a negative focal length is entered to agree with the illustration, then the
image is a virtual image on the same side of the lens as the object and will
give a negative image distance. If a calculation yields a positive focal length,
then the lens is a converging lens rather than the diverging lens in the
illustration. The lens equation can be used to calculate the image distance
for either real or virtual images and for either positive on negative lenses.
The linear magnification relationship allows you to predict the size of the
image.
Virtual Image Formation
A virtual image is formed at the position where the paths of the principal
rays cross when projected backward from their paths beyond the lens.
Although a virtual image does not form a visible projection on a screen, it
is no sense "imaginary", i.e., it has a definite position and size and can be
"seen" or imaged by the eye, camera, or other optical instrument.
A reduced virtual image if formed by a single negative lens regardless of
the object position. An enlarged virtual image can be formed by a positive
lens by placing the object inside the principal focal point.
The Camera and the Eye
Images are formed in a camera by refraction in a manner similar to image
formation in the eye. However, accommodation to image closer objects is
done differently in the eye and camera.
Accommodation in Eye and Camera
Accommodation is the process of adjusting the focus distance of an optical
instrument to the object which is to be viewed. This process is done very
differently by the eye and a camera.
Accommodation from distant to
close objects for the eye is done
by rounding out the lens to
shorten its focal length, since the
image distance to the retina is
essentially fixed.
Accommodation from distant to
close objects for the camera is
done by moving the lens further
from the film since the focal length
is fixed.
aspherical surfaces
mirrors
Note:
An image that is real is always inverted!
An image that is virtual is always upright!
Derivatives of Common Functions
In this table, a is a constant, while u, v, w are functions.
The derivatives are expressed as derivatives with
respect to an arbitrary variable x.