Geometrical Optics 101: Paraxial Ray Tracing Calculations

Ray Tracing Steps | Two Lens System | Ray Tracing Software

Ray tracing is the primary method used by optical engineers to determine optical system performance. Ray tracing is the act of manually tracing a ray of light through a system by calculating the angle of refraction/reflection at each surface. This method is extremely useful in systems with many surfaces, where Gaussian and Newtonian imaging equations are unsuitable given the degree of complexity.

Today, ray tracing software such as ZEMAX® or CODE V® enable optical engineers to quickly simulate the performance of very complicated systems. Paraxial ray tracing involves small ray angles and heights. To understand the basic principles of paraxial ray tracing, consider the necessary calculations and ray tracing tables employed in manually tracing rays of light through a system. This will in turn highlight the usefulness of modern computing software.

PARAXIAL RAY TRACING STEPS: CALCULATING BFL OF A PCX LENS

Paraxial ray tracing by hand is typically done with the aid of a ray tracing sheet (Figure 1). The number of optical lens surfaces is indicated horizontally and the key lens parameters vertically. There are also sections to differentiate the marginal and chief ray. Table 1 explains the key optical lens parameters.

To illustrate the steps in paraxial ray tracing by hand, consider a plano-convex (PCX) lens. For this example, #49-849 25.4mm Diameter x 50.8mm FL lens is used for simplicity. This particular calculation is used to calculate the back focal length (BFL) of the PCX lens, but it should be noted that ray tracing can be used to calculate a wide variety of system parameters ranging from cardinal points to pupil size and location.

Table 1: Optical Lens Parameters for Ray Tracing
Variable	Description
C	Curvature
t	Thickness
n	Index of Refraction
Φ	Surface Power
y	Ray Height
u	Ray Angle

Step 1: Enter Known Values

To begin, enter the known dimensional values of #49-849 into the ray tracing sheet (Figure 2). Surface 0 is the object plane, Surface 1 is the convex surface of the lens, Surface 2 is the plano surface of the lens, and Surface 3 is the image plane (Figure 3).

Remember that the curvature (C) is equivalent to 1 divided by the radius of curvature (R). The first thickness value (t) (25mm in this example) is the distance from the object to the first surface of the lens. This value is arbitrary for incident collimated light (i.e. light parallel to the optical axis of the optical lens). The index of refraction (n) can be approximated as 1 in air and as 1.517 for the N-BK7 substrate of the lens.

In Figure 2, the red box is the value to be calculated because it is the distance from the second surface to the point of focus (BFL). The power (Φ) of the individual surfaces is given by the fourth line and is calculated using Equation 1. Note: A negative sign is added to this line to make further calculations easier. In this example, Surface 1 is the only surface with power as it is the only curved surface in the system.

Step 2: Add a Marginal Ray to the System

The next step is to add a marginal ray to the system. Since the PCX lens is spherical with a constant radius of curvature and a collimated input beam is used, the ray height (y) is arbitrary. To simplify calculations, use a height of 1mm.

A collimated beam also means the initial ray angle (u) is 0 degrees. In the ray tracing sheet, nu is simply the angle of the ray multiplied by the refractive index of that medium. Both variables are included to make subsequent calculations simpler (Figure 4).

Step 3: Calculate BFL with Equations and the Ray Tracing Sheet

Ray tracing involves two primary equations in addition to the one for calculating power. Equations 2 – 3 are necessary for any ray tracing calculations.

where an apostrophe denotes the subsequent surface, angle, thickness, etc. In this example, to find the ray height at Surface 2 (y'), take the ray height at Surface 1 (y) and add it to -0.0197 multiplied by 3.296:

Performing this for ray angle yields the following value. The entire process is repeated until the ray trace is complete (Figure 5).

Now, solve for the BFL by either adjusting the thickness value until the final ray height is 0 (Figure 6) or by backwards calculating the BFL for a ray height of 0. For #49-849, the final BFL value is 47.48mm. This is very close to the 47.50mm listed in the lens' specifications. The difference is attributed to the rounding error of using an index of refraction of 1.517 instead of a slightly more accurate value that was used when the lens was initially designed.