Geometrical Optics

The Meaning of the Symbols

• \(O\): object , \(I\): (virtual) image, \(p\): object distances (positive quantities), \(i\): image distances (negative quantities).
• In the spherical mirrors: \(C\): the center of curvature , \(r\): the radius of curvature, \(F\): the focal point, \(f\) :the focal length.
• \(h\): the height of the object, \(h^{\prime}\): the height of the image. \(m\) is called the \highlight{lateral magnification (横倍率)} produced by the mirror.

Plane Mirrors

• For a point source of light \(O\) (object)

• For an extended object \(O\)

Spherical Mirrors

The Concave Mirror(凹) and the Convex Mirror(凸)

Focal Point of Spherical Mirrors

Mirror Type	Concave	Convex
\(F\) (Focal Point)	real	virtual
\(f\) (Focal Length)	\(f>0\)	\(f<0\)

\[f = \frac{2}{r}\]

Images from Spherical Mirrors

\[\frac{1}{p }+\frac{1 }{i }=\frac{1 }{f }\]

The Principles for Images

• The angles should be small.
• The equation applies to any concave \((f > 0)\), convex \((f < 0)\), or plane \((f = \infty )\) mirror.
• For a convex or plane mirror , only a virtual image can be formed, regardless of the object’s location on the central axis.

Rules to locate an image

• A ray that is initially parallel to the central axis reflects through the focal point F .
• A ray that reflects from the mirror after passing through the focal point emerges parallel to the central axis.
• A ray that reflects from the mirror after passing through the center of curvature C returns along itself.
• A ray that reflects from the mirror at point c is reflected symmetrically about that axis.

\(h\) represents the height of the object, \(h^{\prime}\) represents the height of the image. \(m\) is called the lateral magnification (横倍率) produced by the mirror.

\[m \equiv \frac{h^{\prime}}{h} = -\frac{i}{p}\]

Spherical Refraction

For light rays making only small angles with the central axis, so \(\sin\theta\sim\theta\).

\[\frac{n_1}{p}+\frac{n_2}{i}=\frac{n_2-n_1}{2f}=\frac{n_2-n_1}{r}\]

Thin Lenses

We consider only light rays that make small angles with the central axis (again, they are exaggerated in the figures).

When a thin lens with index of refraction n is surrounded by air, this focal length \(f\) is given by the \highlight{lens maker’s equation}:

\[\frac{1}{f }=(n-1)\left( \frac{1}{r_1}-\frac{1}{r_2} \right)\]

If the lens is surrounded by some medium other than air with index of refraction \(n_{medium}\), we replace \(n\) with \(\frac{n}{n_{medium}}\).

For a thin lens with a focal length \(f\) , \(i\) and \(p\) are related to each other by

\[\frac{1}{p}+\frac{1}{i}=\frac{1}{f}\]

Rules For Thin Lenses

• A ray that is initially parallel to the central axis of the lens will pass through focal point \(F_2\).
• item A ray that initially passes through focal point \(F_1\) will emerge from the lens parallel to the central axis.
• A ray that is initially directed toward the center of the lens will emerge from the lens with no change in its direction because the ray encounters the two sides of the lens where they are almost parallel.