13. Geometrical optics. Photometry. Quantum nature of lightSavchenko Formulas, chapter 13 of 14, 18 formulas
Sections follow the book, and within a section the most used formulas come first. An italic problem number means the formula appears in its statement. Rest the cursor on a number to see the statement.
Paraxial rays parallel to the axis meet after reflection from a spherical mirror at the midpoint of the radius. A concave mirror has a real focus, a convex one a virtual focus, and its $F$ is taken negative. A ray striking far from the axis at an angle $\alpha$ crosses the axis at $R/(2\cos\alpha)$ from the centre, so a wide beam does not meet at one point.
The same thin lens or mirror formula with distances measured from the foci rather than from the lens. Substituting $a = l_1 + f$,$b = l_2 + f$ into $1/a + 1/b = 1/f$ gives $l_1 l_2 = f^2$ at once.
The profile that brings every ray parallel to the axis to one point is a parabola with its focus there. It follows from equal optical paths, the distance from the wave front to a point of the profile plus the distance from it to the focus is the same for every ray.
A diverging lens has $f < 0$, a virtual image has $b < 0$, a virtual object (a beam converging onto the lens) has $a < 0$. With the same signs the formula holds for a spherical mirror with $f = R/2$. The image formed by the first lens is the object of the second.
Angles are measured from the normal to the boundary. The product $n\sin\alpha$ is conserved along a ray through any stack of plane layers, so in a layered medium only the indices at the start and the end matter. For small angles the sines are replaced by the angles.
The two surfaces add, each gives $(n/n_0 - 1)/R$. A concave surface has a negative radius, a flat one an infinite radius. In water ($n_0 > 1$) a glass lens loses power, and an air lens in water diverges where a glass one converges.
A ray coming from the denser medium at more than the critical angle does not leave and is reflected entirely. Towards air $\sin\alpha_{\text{cr}} = 1/n$. The condition for a ray to leave a prism or a fibre is written as $\sin\gamma < 1/n$ for the angle at the inner face.
Seen nearly along the normal, an object in a medium of index $n$ appears $n$ times closer to the boundary. This is the small angle refraction law, $n\beta = \alpha$, applied to two rays from one point. For an observer inside the medium an object outside appears $n$ times farther.
For a small wedge angle and nearly normal incidence the deviation does not depend on the angle of incidence. Between media of indices $n_1$ and $n_2$ the deviation is $(n_1 - n_2)\alpha$, and in the focal plane of a lens it shifts the image by $f\delta$.
distances of the object and the image from the lens
f
focal length
The lateral magnification is the ratio of distances because the ray through the centre of the lens goes straight. A small length along the axis is magnified $\Gamma^2$ times, so the image of a cube is stretched along the axis.
The optical powers of thin lenses put together without a gap add. So do the powers of refracting surfaces and a mirror in a lens with a silvered face, where the light crosses the lens twice.
A magnifier lets the object come closer to the eye than $d_0$ and still be sharp. An object in the focal plane gives a parallel beam, the angle $h/F$ instead of $h/d_0$.
A lens collecting light from an extended source takes a flux proportional to the area of its aperture and to the solid angle the source subtends. The illuminance of the image therefore grows as the square of the lens diameter over the image distance and does not depend on the distance to a far object.
$$E = \frac{I\cos\varphi}{r^2}, \qquad I = \frac{P}{4\pi}$$
I
luminous intensity of the source
r
distance to the surface
$\varphi$
angle between the ray and the normal
P
total flux of an isotropic source
The flux into a solid angle $d\Omega$ is $I\,d\Omega$, and a patch $dS$ at distance $r$ subtends $dS\cos\varphi/r^2$. Hence the inverse square law and the cosine factor. Illuminances from several sources add.
Each thin layer absorbs the same fraction of the light that reaches it, $dI/I = -dx/L$, hence the exponential. For a thin layer $I \approx I_0(1 - x/L)$.
Light is absorbed and emitted in quanta. A photon carries momentum, so an atom absorbing or emitting one recoils at $h\nu/mc$, and a mirror reflecting it receives twice its momentum. A collision of a photon with a particle is solved with energy and momentum conservation, as an elastic impact.
A ray passing at a distance $b$ from a mass $M$ turns by a small angle $\alpha$. A ray passing through an extended body feels the mass inside a cylinder of radius $b$, so the body acts as a gravitational lens.
A photon of energy $h\nu$ behaves as a mass $h\nu/c^2$ and, climbing out of the gravitational field, loses $GMh\nu/(Rc^2)$ of energy. The frequency at infinity is therefore lower.