Theories & Formulas

Technical note, theories and formulas on fundamental properties within an optical system.

Sign Conventions

Before proceeding with the methods of analysis for an imaging system, a few sign conventions must be established. The first is that light travels from left to right. See Sign Conventions Table below.

Parameter Parameter is Positive (+) if the following is true: Parameter is Negative (-) if the following is true:
Radius of curvature Center of curvature is to the right of the vertex Center of curvature is to the left of the vertex
Object distance Object is to the left of the lens Object is to the right of the lens
Image distance Image is to the right of the lens Image is to the left of the lens
Back focal length Back focal point is to the right of the last element Back focal point is to the left of the last element
Front focal length Front focal point is to the left of the first element Front focal point is to the right of the first element

Ray Tracing

By propagating the rays linearly through homogeneous media and applying Snell’s Law at the interfaces between media, light rays can be traced through an optical system. Such a system may consist of various optical components (lenses, prisms, mirrors, etc.). Each optical component can be treated as a series of individual interfaces and transfers. For example, a simple lens is a piece of transparent material with two surfaces of specified radii and a thickness. The ray trace through the lens consists of three segments: refraction at the first surface, propagation through the lens, and refraction at the second surface. For a specific optical system, this series of three simple operations can be repeated to trace the rays through the entire system.

Paraxial Analysis

An initial analysis of an optical design is often performed using the paraxial approximation, which is valid at the small angles of incidence common for rays propagating near the optical axis. Using the series expansion of the sine function given by

sin θ = tan θ – θ³/3! + θ5/5!
(θ measured in radians),

paraxial (first-order) theory retains only the first term. In the following calculation, assume sin θ = tan θ = θ, and cos θ =1.

Snell’s Law then can be written as: n1θ1 = n2θ2.

Focal Length

Rays that pass through either focal point are parallel to the optical axis on the other side of the lens. The focal length is related to the object and image locations by the “lens formula”:

1/f = 1/S +1/S¹

Where:

S = object distance
S¹ = image distance

Magnification

The magnification, M, is defined as the ratio of the image height, y¹, to the object height, y:

M = y¹/y; also, M = S¹/S.

Thin Lens Theory

The simplest case is that of a “thin” lens of zero thickness in air. All of the refraction occurs at the plane of the lens, and all distances can be measured from this plane, as shown in the figure below. The lens is described by:

1/f = (n-1) (1/R1 – 1/R2)

Where:
f = focal length
R1 = radius of curvature of the first surface of the lens
R2 = radius of curvature of the second surface of the lens
n= index of refraction of the material of the lens.

Thin Lens Imaging

Thin Lens Imaging

Thick Lens Theory

While a thin lens is a convenient idealization, a physical lens possesses a finite thickness. Accounting for this leads to the “lensmaker’s formula” for a thick lens in air:

Thin Lens Imaging Figure

Where:
t = center thickness of the lens
f = effective focal length (EFL)
R1 = first radius of curvature
R2 = second radius of curvature
n = index of refraction of the lens

Principal Planes

With a thick lens, the concept of principal surfaces is introduced. A light ray traced from the object, through the first focal point (F1), consists of three segments, as seen in the Thick Lens Imaging figure below: the first, in air, from the object point to the lens; the second, within the lens; and the third, in air, after the lens and parallel to the optical axis. If the entering and emerging rays are extended, they will intersect in a locus of points that forms the primary (or front) principal surface (H1). Near the axis, this surface becomes nearly flat and commonly is called the front principal plane. In a similar manner, rays incident parallel to the axis of the lens exit through the back principal plane (H2) and intersect the back focal point (F2). The secondary (or back) principal plane is defined by the intersection of these incident and emergent rays. The principal planes represent a single imaginary surface of refraction that replaces the real surfaces of the lens. Although H1 and H2 are located at different positions within the system, this separation is “invisible” to the ray. For the purpose of ray tracing, a ray of height yh at H1 can be directly translated to H2; its height remains yh. Also, when the lens is used at unit magnification (1×) the principal planes are conjugate to each other.

Principal Planes, Thick Lens Imaging diagram

The principal planes serve as the references for the location of the front focal point, back focal point, object and image positions. Both the lensmaker’s and the lens formulae involve the effective focal length f, which is measured from H1 and H2. While the principal planes are useful in lens design calculations, their usefulness in a physical setup is limited because either plane may lie inside or outside the lens itself. In a physical setup, the front focal length (FF) and the back focal length (BF) are more useful than the principal planes. The front focal length is measured from the front focal point to the vertex of the first surface of the lens; the back focal length is measured from the vertex of the last surface of the lens to the back focal point.

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