# Is there a more accurate form of the mirror equation $\frac{1}{f}=\frac{1}{u} + \frac{1}{v}$?

In the mirror equation $$\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$$

Q1: Are $u$ and $v$ the distances from the object to the mirror surface or the distance from the object along the principal axis to the pole?

These distances differ by a very small amount but they exist all the same. I am familiar with the usual derivation of the formula using a diagram, alternate angles, an isosceles triangle and then a small angle approximation.

Q2: Is there a formula that doesn't make any approximations?

• The gaussian form is $(u-f)(v-f)=f^2$ but is really the same – Narasimham Jun 29 '17 at 18:36

There is more than one approximation here. Some apply to mirrors and lenses. Some apply to the propagation of light in general. Lenses are more common that curved mirrors. Much more attention has been paid to lens design than mirror design. But the techniques are similar.

First, the small angle approximation assumes angles are so small that it doesn't matter which of those distances you use. It works for paraxial rays, which are very close the principal axis. These are the rays that define the focal length. So $1/f = 1/u + 1/v$ is correct as is.

For non-paraxial rays, you need to trace the path of the ray through the system. Whenever the ray hits the surface of a lens or mirror, you calculate the location of where it hits and the new direction of the ray.

Complex lenses have been designed for longer than computers have been available. If you are tracing rays through a system manually, you traces as few rays as possible, and you use the simplest approximation that is sufficiently accurate.

Calculations can be simplified with the thin lens approximation. The focal length of a lens can be calculated from the radii of the surfaces, the index of refraction, and the thickness at the center. For many lenses, the thickness has a small effect and can be ignored. The lens is treated as a plane. u and v are measured parallel to the principal axis to the plane.

For more accuracy, thickness does matter. u and v are measured parallel to the axis to specific points within the lens. The points are on the principal planes of the lens.

For ray tracing, the small angle approximation isn't good enough. Before computers, people often used a power series.

$$sin(\theta) = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - ...$$

The approximation

$$sin(\theta) = \theta - \frac{\theta^3}{3!}$$

was accurate enough for many purposes. Using this, you find that the paraxial rays behave as the simpler approximation predicts, but many other rays are not focused to that point. These failures to focus to a point are called lens aberrations.

Using this approximation, people identified what are called 3rd order aberrations. For example, spherical aberration is the failure of rays parallel to the axis to be focused to a point by a lens or mirror with spherical surfaces. A parabola would have no spherical aberration. Coma is the failure for rays at an angle to meet at a point.

Light has a mix of wavelengths. The index of refraction of glass and other materials varies with wavelength. This means that $1/f = 1/u + 1/v$, but a different f must be used for each wavelength. Or, given a fixed u, v is different for each wavelength. This is called chromatic aberration. Mirrors do not have this aberration.

Modern lens design programs do not need to use a third order approximation to $sin(\theta)$. Never the less, they must report the aberrations in their traditional forms.

The topic of predicting light propagation with rays is called geometric optics. For complex lenses with multiple elements, materials can be chosen to minimize chromatic aberration, and curves can be chosen to minimize other aberrations.

But light is a wave. Waves diffract. If light passes through a pinhole, you get a central bright spot surrounded by a series of bright and dark rings. A lens or mirror can be thought of as a large pinhole. Even when geometric optics predicts that all rays are focused to a perfect point, diffraction means it will really be a spot. The size of the spot is determined by the diameter of the lens or mirror. For a circular lens in air, the diameter of the spot is given by

$$d_{spot} = 1.22 \lambda f/d_{lens}$$

For a camera, this limits the sharpness of the image. For a telescope or microscope, it limits the separation at which two objects can be resolved. For a laser, it limits the intensity of the focal spot.

Optical systems where ray aberrations are no larger than diffraction are said to be diffraction limited.

This is enough for most purposes. But if you want no approximations, there are more things to consider. It all depends on what you are interested in and how accurate you want to be.

For example, polarization sometimes matters. A beam splitter is a partially reflecting mirror. It is usually mounted at a 45 degree angle, so that half of a laser beam is reflected sideways, and half is transmitted. The two beams are polarized.

Absorption sometimes matters. A typical high power CO2 industrial laser has 100 Watts of power in the beam. If 0.1% is absorbed by a lens or mirror surface, the lens or mirror will heat up. This can change the shape and defocus the beam. Or it can crack a lens. Copper mirrors are sometimes used instead of lenses because they are easier to cool.

Likewise good anti reflection coatings are needed. Unexpected reflections can come to a focus and start a fire.

Air scatters light in a wavelength dependent way. This is a small effect, but over long distances it makes the sky blue. LIGO is a large, extremely precise optical interferometer that recently detected gravitational waves. It has laser beam paths 4 km long. Scattering is one reason that the beams are in high vacuum.

When light is hits a moving object, the object sees a different wavelength. This is called the Doppler shift. Light reflected from a moving mirror has a different wavelength that a stationary mirror. Air molcules bouncing off the mirrors in LIGO would move the mirrors too much. This is another reason the beams are in vacuum.

The point is that "no approximations" means more than you might think.

There is a more general (exact) formula for a spherical mirror. This formula was discovered by H. A. Elagha and was published in the journal of the optical society of america in 2012. The paper has the title: "Exact ray tracing formulas based on a nontrigonometric alternative to Snell's law" . This formula has the form:

where So and S1 are the the distances of the object and the image from the mirror vertex respectively. h is the hieght of the point of incidence at the mirror and R is the radius of curvature.[![enter image description here][2]][2]

• Please use MathJax in this site. – exp ikx Mar 18 at 14:34