Propagation of light

Visible light is a narrow part of the electromagnetic spectrum and in a vacuum all electromagnetic radiation travels at the speed of light:

The above number is now accepted as a standard value and the value of the meter is defined to be consistent with it. In a material medium the effective speed of light is slower and is usually stated in terms of the index of refraction of the medium. Light propagation is affected by the phenomena refraction, reflection, diffraction, and interference. The behavior of light in optical systems will be characterized in terms of its vergence.

Refraction of Light

Refraction is the bending of a wave when it enters a medium where it's speed is different. The refraction of light when it passes from a fast medium to a slow medium bends the light ray toward the normal to the boundary between the two media. The amount of bending depends on the indices of refraction of the two media and is described quantitatively by Snell's Law.

Refraction is responsible for image formation by lenses and the eye. As the speed of light is reduced in the slower medium, the wavelength is shortened proportionately. The frequency is unchanged; it is a characteristic of the source of the light and unaffected by medium changes.

Snell's Law

Snell's Law relates the indices of refraction n of the two media to the directions of propagation in terms of the angles to the normal. Snell's law can be derived from Fermat's Principle or from the Fresnel Equations.

Index of Refraction

The index of refraction is defined as the speed of light in vacuum divided by the speed of light in the medium.

The indices of refraction of some common substances are given below with a more complete description of the indices for optical glasses given elsewhere. The values given are approximate and do not account for the small variation of index with light wavelength which is called dispersion.

Fermat's Principle and Refraction

Fermat's Principle: Light follows the path of least time. Snell's Law can be derived from this by setting the derivative of the time =0. It shows how the refraction of light at an interface depends upon the indices of refraction.

This derivation makes use of the calculus of maximum-minimum determination and the definitions of the triangle trig functions.

Lenses

Thin Lens Equation

A common Gaussian form of the lens equation is shown below. This is the form used in most introductory textbooks. A form using the Cartesian sign convention is often used in more advanced texts because of advantages with multiple-lens systems and more complex optical instruments. Either form can be used with positive or negative lenses and predicts the formation of both real and virtual images.  Does  not apply to thick lenses.

Lens-Maker's Formula

For a thin lens, the power is approximately the sum of the surface powers.

The radii of curvature here are measured according to the Cartesian sign convention. For a double convex lens the radius R1 is positive since it is measured from the front surface and extends right to the center of curvature. The radius R2 is negative since it extends left from the second surface.

Thick Lens Imaging

The thin lens equation cannot be used directly to find images formed by thick lenses. Even when Gullstrand's equation is used to find the equivalent power of the thick lens, that power is with respect to a principal plane. The thickness of the lens alters the image position and the center of the thick lens cannot in general be used as the lens position. Three methods are presented for finding the image formed by a thick lens.

Gullstrand's Equation

Thick lenses can be handled with thin lens type equations if the distances are measured from hypothetical principal planes. The power of a lens with respect to the second principal plane H2 is given by Gullstrand's Equation:

The focal length f with respect to that second principal plane is given by:

Forms of Gullstrand's Equation are applicable to both separated lenses and to single thick lenses. Since Gullstrand powers and focal lengths are measured with respect to a hypothetical plane, it is often more useful to deal with front and back vertex focal lengths.

Surface Power

The surface power of a lens can be constructed geometrically as illustrated above. It depends upon the indices of refraction and the radius of curvature of the surface. The power of a thin lens is approximately the sum of the surface powers of its surfaces. For a thicker lens, the surface powers can be used in Gullstrand's equation. The expression for surface power obtained above is only valid for light rays at small angles where the angle in radians is approximately equal to the sine and the tangent of the angle. These rays are called paraxial rays. Most of the common expressions in geometrical optics are valid only in this paraxial approximation. Note that the distance d in the illustration above is left out of the expressions since for small angles it becomes very small.

The surface power for a convex surface on the front vertex of a lens is positive because n(lens)>n(air). If the back surface of a lens is convex, it also has a positive power since by the Cartesian sign convention the radius is negative. For a thin lens, the power of the lens is approximately the sum of the surface powers. The relationship for lens power is called the lens maker’s formula.

Vergence

The vergence of light is defined by
Vergence = V = n/L
where n is the index of refraction of the medium and L is the distance in accordance with the Cartesian sign convention.

Since the distance L1 is measured from the wavefront and light is traveling left to right, it is a negative distance and the vergence is negative (divergent). L2 is positive since it is directed to the right from the wavefront (convergent).

The change in vergence when the light encounters a refracting surface is equal to the power of the surface Ps:
V + Ps = V'

System Matrix

For systems of multiple thick lenses, it is sometimes useful to represent the system by a system matrix. The matrix is built up by multiplying the refraction matrices and translation matrices. The positions of the principal planes, the front and back surface powers, and the equivalent focal length of Gullstrand's equation can be calculated from the system matrix.

Optical Glasses

The most common types of glasses used in optics are crown glasses and flint glasses, designations based on their dispersions. Flint glasses contain lead. These designations are further subdivided by composition and have letter designations and number designations called "glass numbers".

Common crown glasses have indices of refraction around 1.5-1.6, while extra dense flint glass may have an index as high as 1.75. Lenses of crown and flint glasses are often used in multi-component lenses because of their complementary properties. For example, a strong positive crown lens with its low dispersion may be used in a doublet with a weaker negative lens of flint glass (high dispersion) to correct for chromatic aberration.
The design of multi-component lenses requires very exacting specifications for the glasses used. Professional optics books have detailed tables of glasses with their glass numbers, densities, softening temperatures, etc. For example, Table 11.6a in Waynant & Ediger.

Aberrations

In an ideal optical system, all rays of light from a point in the object plane would converge to the same point in the image plane, forming a clear image. The influences which cause different rays to converge to different points are called aberrations.