Optics

Reflection and refraction

Reflection

When a ray hits a reflective surface, the angle of incidence to the normal is equal to the angle of reflection. Some energy is lost to the surface when the ray reflects.

\theta_\text{incidence} = \theta_\text{reflection}

Refraction

The angle of incidence \theta_1 of a ray hitting a boundary between two materials with different optical densities (n_1 and n_2) determines the angle of refraction \theta_2. This formula is called Snell’s law.

n_1 \sin\theta_1 = n_2 \sin\theta_2

When a ray travels from a less dense to a more dense material (n_1 < n_2), the ray bends towards the normal. When a ray travels from a more dense to a less dense material (n_1 > n_2), it bends away from the normal. In practice, a portion of light will also reflect off the boundary, and different wavelengths of light will have different IORs in the same medium (with higher wavelengths bending more than lower wavelengths).

The index of refraction n of a material can be calculated from the speed of light c in a vacuum and the speed of light v in the material. c is always 3.0 \times 10^8 \unit{ms}^{-1}.

n = \frac{c}{v}

The IOR of a vacuum is always 1, by definition. Air is close enough to 1 to count. Water is 1.33, glass is 1.5 or so.

Total internal reflection

When a ray travels from a more dense to a less dense material, the angle of refraction can be greater than 90°, meaning that the ray refracts off the boundary and back into the original material. The minimum angle of incidence \theta_1 required to achieve an angle of refraction of 90° can be calculated from Snell’s law.

n_1 \sin\theta_1 = n_2 \sin 90°
\theta_1 = \sin^{-1} \left( \frac{n_2}{n_1} \right)

Mirrors and lenses

Mirrors

A concave mirror is converging, and a convex mirror is diverging.

The features of a mirror are:

  • The center of curvature, the point equidistant from all points on the mirror. This distance is called the radius r.
  • The focus, given by f = \frac{r}{2}. The focus for a converging mirror is positive (in front of the mirror), and a diverging mirror is negative (behind the mirror).
  • All rays parallel to the axis reflect through the focus.

The principle rays for a mirror are:

  • Parallel to the axis from the object, then reflected through the focus.
  • Through the focus from the object, then reflected parallel to the axis.
  • Through the center of curvature from the object, then reflected exactly 180° back.
  • To the pole from the object, then reflected such that \theta_i = \theta_r.

Lenses

A convex lens is converging, and a concave lens is diverging.

The features of a lens are:

  • The focus, which is positive for a converging lens (through the lens) and negative for a diverging lens (behind the lens).
  • The optical center, at the intersection of the axis and the lens plane.

The principle rays for a lens are:

  • Parallel to the axis from the object, then refracted through the focus.
  • Through the focus from the object, then refracted parallel to the axis.
  • Through the optical center from the object, passing straight through without refracting.

Image features

  • An image is upright if the image is on the same side of the axis as the object, inverted otherwise.
  • An image is enlarged if the distance d_i between the image and the plane is greater than the distance d_o between the object and the plane, diminished otherwise.
  • An image is real if it’s through the lens or in front of the mirror (d_i > 0), or virtual if the image is behind the lens or mirror.

Image formula

The distance d_i from the plane to the image, the distance d_o from the plane to the object, and the distance f from the plane to the focus (focal length) are related:

\frac{1}{d_i} + \frac{1}{d_o} = \frac{1}{f}

If f < 0, the mirror or lens is diverging.

Magnification formula

The magnification M is related to the height h_i of the image from the axis and the height h_o of the object from the axis, and also to d_i and d_o:

M = \frac{h_i}{h_o} = \frac{-d_i}{d_o}

If M < 0 then the image is inverted, and if \|M\| > 1 then the image is enlarged.