# Optical invariants

Optical invariants are mathematical expressions which represent the product of several optical parameters . The value of multiplication for each optical invariant is constant despite the fact that each optical parameter changes from one part (or element) of the optical system to another.

Optical invariants have important significance during optical systems design because they make it possible achieve quicker results and avoid mistakes in calculations of optical parameters.

There are 4 main invariants of optical systems:

Lagrange-Helmholtz invariant for paraxial optics;

Abbe sine condition for high aperture systems;

Straubel invariant for illumination optical systems with light pencils transformation;

Etendue invariant for illumination optical systems with extended source.

## Lagrange-Helmholtz invariant

Fig 1 presents an schematic optical scheme which functioning is shown with help of principal planes ℎ and ℎ′.

Symbols on the scheme:

𝑦 and 𝑦′ are heights of object and image;

𝛼 and 𝛼’ aperture angles in object and image spaces;

𝑛 and 𝑛’ are indexes of refraction in object and image spaces.

In case of small angles and heights of rays relation between these parameters is represented as shown below.

This is well known invariant of Lagrange-Helmholtz. It shows that product of aperture angle value and height of field is constant in image space and object space. This relation works for any optical system with any number of optical components.

Invariant makes it possible to define relation between image aperture angle and height of the image when object aperture angle and its height are known and fixed. Then main optical parameters of the optical system can be defined.

In an existing optical system, Lagrange-Helmholtz make it possible to estimate relation between two pairs parameters of object and image.

**Modifications of Lagrange-Helmholtz invariant for different methods of object to image conjugation**

**Infinite to infinite conjugation**

Example: Galilean telescope system

**Infinite to finite conjugation**

Example: Objective lens of camera

Where ω is field angle, D0/2 is half of aperture diameter, F is f-number.

### Lagrange-Helmholtz invariant application

Lagrange-Helmholtz invariant is very important for estimating the relation of parameters of systems in paraxial region, where apertures and fields of view are quite small.

The invariant makes it possible to define main parameters of paraxial optical systems - focal lengths, magnification, positions of object and image, entrance pupil and exit pupil, principal planes and etc.

## Abbe sine condition

Lagrange-Helmholtz invariant is true for optical system with relatively small of y (ω) and α. For optical system with essential α similar to microscope lens more accurate invariant is needed. Its name is Abbe sine condition.

Comparison of Lagrange-Helmholtz invariant with Abbe sine condition shows that Sine condition uses sinus of aperture angles instead of values of aperture angles.

### Abbe sine condition application

Sine condition is true for optical systems which don’t have spherical aberrations and coma. And vice versa, if sine condition is true for given optical system that optical system doesn’t have spherical aberration and coma. Such optical system named as aplanatic system. In other words sine condition is condition of aplanatism. This is a key indicator of microscope lens performance.

Sometimes if optical system has residual aberrations this can be characterized as OSC (offence against the sine condition). Level of deviation from sine condition shows on extent of proximity of concerned optical system to aplanatic optical system.

One of result from sine condition is that maximal aperture aplanatic systems is 0.5 although practically it isn’t achieved.

## Light pencil invariant and Straubel theorem for transferring

Let’s consider propagation of light pencil beam through the refractive surface (Fig. 5). The volume of light pencil is a volume limited by surfaces 𝑑𝑆1, 𝑑𝑆 and marginal rays between them. Two light pencils beams are presented on Fig. 5:

Between 𝑑𝑆1 surface and refractive surface 𝑑𝑆;

Between refractive surface 𝑑𝑆 and surface 𝑑𝑆2.

Rays which propagate between surfaces form several solid angles: 𝑑Ω, 𝑑Ω1 and 𝑑Ω2. Light pencil invariant demonstrates interrelation of these angles and surfaces:

Straubel theorem takes into account refraction coefficient of media with different indexes refraction 𝑛 and 𝑛’.

This equality is valid for any number reflections or (and) refractions.

## Etendue invariant

Real optical system works with extended sources with square 𝑆 and finite apertures determined by sizes and construction of this system. Area of light spot generated by rays passed through optical system is determined by Etendue invariant:

where S and S’ are areas of source and its image and Ω is solid angle which is defined like this:

where 𝑢 is aperture angle.

Subject to this, Etendue invariant looks like this:

### Etendue invariant application

**The limit of light spot size**

We can use Etendue invariant to find the ratio of source area and area of light spot formed with help of optics.

Maximum value of sine function for image aperture angle is 1 when angle 𝑢’=90°. In this case the ratio will have maximum value (𝐶𝑚𝑎𝑥). It is defined by this formula:

**Throughput of LED illumination optical system**

If the optics are not large enough to overcome the Etendue of the source, then there will be an additional efficiency loss. In this case Etendue of the source will be limited by Etendue of the optics and Etendue invariant will not work. But it can help to find level of losses and even optimize them with help of formula provided below.