The law of refraction (Snell's Law): When a ray of light is refracted at an interface between two uniform media, the transmitted ray remains in the plane of incidence and the sine of the angle of refraction is directly proportional to the sine of the angle of incidence.
The law of reflection: When a ray of light is reflected at an interface dividing two uniform media, the reflected ray remains in the plane of incidence, and the angle of reflection equals the angle of incidence. The plane of incidence is defined by the incident ray and the surface normal vector at the point of incidence.
Snell's Law is usually written:
with
Using these two laws we can analyze systems of lenses and mirrors like the all-lens system below:
After all this, we'd like to use Snell's Law in our geometrical optics approximation
to reality. Linearize problem with
paraxial rays, those that never
make
large angles with the optical axis. I.e. assume
The paraxial Snell's Law is
This is
a "small angle
approximation".
Suc
linear approximations describe a given function
(in some locality) as a linear function. Infinite series are
in your future!
Here,
Different rays travel different distances through lenses, which are fatter and thinner. So just what happens depends on just where and at what angle the ray enters the lens. Inconvenient, nonlinear. The thin lens assumption is: "consider an infinitely thin lens" with all the refractive power and none of the annoying size of an actual object.
Geometrical optics uses pretty simple algebra (but pretty complicated diagrams) to derive elegant formulae (in terms of things like radii of curvature, indices of refraction, and distances) that describe spherical mirrors, refraction at spherical surfaces, thin lenses, thick lenses etc.
Focal length f is the (signed) distance to the image it forms of an object at infinity. Convex lenses have positive focal lengths, Concave negative).
The lensmaker's equation predicts the focal length of a lens
in terms of its refractive index, that of the medium it is in, and
its radii of curvature. In air (of refractive index
The power P of a lens measures
how strongly it bends light, and is defined as
We measure angles that rays deviate from the optical system
(
if the ray impinges on the lens at height
Diagram the system as thin lenses, a 2-D plot of Y versus X, with the X axis
being the optical axis.
An object being imaged is considered to be in an input plane at
the
left of the diagram (below, it's at
Consider a ray
|y_{1}| = |1 L| |y_{0}| |θ_{1}| = |0 1| |θ_{0}|
This simple matrix is how we describe the change in the ray as if moves through some uniform medium for an axial distance.
Remember
So...
|y_{1}| = |1 0| |y_{0}| |θ_{1}| = |-1/f 1| |θ_{0}|
The transfer matrices for spherical or refraction interfaces, spherical mirrors, and thick lenses are similarly simple.
Express a paraxial system with elements described by 2x2 matrices,
say M1, M2, M3, M4 in order from left to right. Then for ray
Note the system components appear in reverse order.
Use ray-transfer technique find: focal points, nodal points and first and second principal planes.
The four elements of the transfer matrix have discernable semantics, which can be illuminated by considering the physical meaning of setting each one to zero. For now, though, we're done.
Consider three lenses of focal length 200, -50, and 50 mm. The first and second are separated by 100 mm. the second and third by 50 mm. The input plane is 200 mm. in front of the first lens. What is the (back, effective) focal length of the system?
Send a ray parallel to the axis into the 6-element system
(translation, lens, translation, lens, translation, lens). The last
ray we get is the one emerging from the last lens at X=350. It turns out
to be (3, -.04)^{T}. We're after the focal length
The axial image point of a system is where the image of a ray starting out at the origin crosses the optic axis. It's like the focal point, and calculated the same way, only using a different initial ray. It turns out that all the system's image points from an object at the origin will fall in the plane at that distance, so that is where an in-focus image will be formed.
Use slightly different 3-lens system: lenses are spaced out by 300, 100, and 50 of focal lengths 200, -50, and 50. Shooting out a ray from the origin at angle .01 and calculating the axial image distance (it's 95 mm. out from the third lens) we can compose a final translation after the 3rd lens of 95 mm to get this plot:
Several Rays:
Use the y height at the image plane to get the ratio object-height/image-height, or linear magnification, as .2 (the image height is upside-down).
Ray-casting: given a point of view and direction of gaze (ray), compute the color and intensity of the light in that direction. In geometrical optics we follow light-paths (rays) through optical systems with geometry, some algebra, and a few physical laws.
Basic case: given a 3-D point of origin for the ray and its direction, where does it intersect a given plane in space? No different from the high-school algebra "line-intersect plane" problem, and in optics sometimes called "intersecting a ray with a plane mirror".
As usual, represent points p, x, r etc.in 3-D by
The standard way to intersect a vector with something is to stretch it out
in its direction until it hits; the resulting length is all we need.
So: Any point on a ray can be written
Ray:
with
The plane equation is linear, and a 3-D version of the familiar line
equation:
Plane:
Written like this,
Force our
infinite mirror, or plane, to pass through the origin, so we can
describe it with a linear equation: one strictly in
Plane through Origin:
For a more general raycasting project, still pretty easy, see Pinholes and Beachballs (or spherical chickens!).
The equations Ray and Plane above are four linear equations. A
solution to them gives the
Ray origin:
Ray direction:
3 Ray Equations, 1 Plane Equation:
We need four equations for unknowns
Express above system as
matrix equation:
If
Then ignore