# Geometric Optics

Reflection or refraction at a plane surface:  When rays diverge from an object point $P$ and are reflected or refracted, the directions of the outgoing rays are the same as though they had diverged from a point $P'$ called the image point.  If they actually converge at $P'$ and diverge again beyond it, $P'$ is a real image of $P$; if they only appear to have diverged from $P'$, it is a virtual image.  Images can be either erect or inverted.

Lateral magnification:  The lateral magnification $m$ in any reflecting or refracting situation is defined as the ratio of image height $y'$ to object height $y$, $m=\frac{y'}{y}$.  When $m$ is positive, the image is erect; when $m$ is negative, the image is inverted.

Focal point and focal length:  The focal point of a mirror is the point where parallel rays converge after reflection from a concave mirror, or the point from which they appear to diverge after reflection from a convex mirror.  Rays diverging from a focal point of a concave mirror are parallel after reflection; rays converging toward the focal point of a convex mirror are parallel after reflection.  The distance from the focal point to the vertex is called the focal length, denoted as $f$.  The focal points of a lens are defined similarly.

Relating object and image distances:
Plane mirror: $s=-s', m=1$.
Spherical mirror: $\frac1s+\frac1{s'}=\frac2R=\frac1f, m=-\frac{s'}s$.
Plane refracting surface: $\frac{n_a}s+\frac{n_b}{s'}=0, m=\frac{n_as'}{n_bs}=1$,
Spherical refracting surface: $\frac{n_a}s+\frac{n_b}{s'}=\frac{n_b-n_a}R, m=\frac{n_as'}{n_bs}$.

These object–image relationships are valid only for rays close to and nearly parallel to the optical axis; these are called paraxial rays.  Non-paraxial rays do not converge precisely to an image point.  This effect is called spherical aberration.

Thin lenses:  The lensmaker’s equation relates the focal length of a lens to its index of refraction and the radii of curvature of its surfaces.  $\frac1s+\frac1{s'}=\frac1f$, $\frac1f=(n-1)(\frac1{R_1}-\frac1{R_2})$.

Sign rules:  The following sign rules are used with all plane and spherical reflecting and refracting surfaces:

• $s>0$ when the object is on the incoming side of the surface (a real object); $s<0$ otherwise.
• $s'>0$ when the image is on the outgoing side of the surface (a real image); $s'<0$ otherwise.
• $R>0$ when the center of curvature is on the outgoing side of the surface; $R<0$ otherwise.
• $m>0$ when the image is erect; $m<0$ when inverted.

Cameras:  A camera forms a real, inverted, reduced image of the object being photographed on a light-sensitive surface.  The amount of light striking this surface is controlled by the shutter speed and the aperture.  The intensity of this light is inversely proportional to the square of the $f$-number of the lens: $f$-number$=\frac{\mathrm{Focal\ length}}{\mathrm{Aperture\ diameter}}=\frac{f}{D}$.

The eye:  In the eye, refraction at the surface of the cornea forms a real image on the retina.  Adjustment for various object distances is made by squeezing the lens, thereby making it bulge and decreasing its focal length.  A nearsighted eye is too long for its lens; a farsighted eye is too short.  The power of a corrective lens, in diopters, is the reciprocal of the focal length in meters.

The simple magnifier:  The simple magnifier creates a virtual image whose angular size $\theta'$ is larger than the angular size $\theta$ of the object itself at a distance of 25 cm, the nominal closest distance for comfortable viewing.  The angular magnification $M$ of a simple magnifier is the ratio of the angular size of the virtual image to that of the object at this distance.  $M=\frac{\theta'}{\theta}=\frac{25\ \mathrm{cm}}f$.

Microscopes and telescopes:  In a compound microscope, the objective lens forms a first image in the barrel of the instrument, and the eyepiece forms a final virtual image, often at infinity, of the first image.  The telescope operates on the same principle, but the object is far away.  In a reflecting telescope, the objective lens is replaced by a concave mirror, which eliminates chromatic aberrations.