Ray Tracing of Thick Lenses

By

Brent McCardle

The purpose of this article is to build on what was discussed inRay Tracing of Thin Lenses. If you have not read that article it would be wise to do so. This article assumes a basic understanding of mathematical concepts and an understanding of some optics.

Tracing a thick lens is different because the ray of light is going to be refracted at more than just one meridian. A Thick Lens is defined as a lens in which the separation between the two surfaces is too great to be ignored. For example the lens power measured from the front of the lens will be different than if it were measured from the back, due to the thickness.

A couple of points to note, all rays that are traced will be paraxial, close to the axis. Usually opticians tend to give a positive radius to the front curve and a negative radius to the back curve, because the powers are positive and negative respectively. We need to understand that a positive radius is defined as the center of curvature being to the right of the lens and a negative radius is when the center of curvature is to the left.

Since we are dealing with thick lenses we need to know where to measure the focal length, object distance, and image distance. We will not measure these items from the center of the lens or from the lens vertices's, but rather from the principal planes (). Principal planes are where refraction is assumed to occur.

Fig.1

Now we need to find the equivalent power of the lens. The equivalent power is assigning one power to a two lens system. For example, if you read a thick lens in the focimeter on the front and then read the back there would be two different powers. The equivalent power is the position of the principal planes with respect to the first and second focal points. Once we find the equivalent power we can then find the principal planes.

We will explain in more detail later, just understand that is measured from the front vertex and is measured from the back vertex. Also understand that and are measured from and respectively.

Formulas to trace a thick lens

Radius of Curvature

is the radius of curvature in Meters

is the refractive index of the the refracting side of the surface

is the refractive index of the incident side of the surface

D is the power of the surface in Diopters

Front Vertex Power

is the front vertex power

is the power of the front of the lens

is the power of the back of the lens

is the thickness in Meters

is the refractive

Back Vertex Power

is the back vertex power

is the power of the front of the lens

is the power of the back of the lens

is the thickness measured in Meters

is the refractive index

Primary Focal Point ()

is the primary focal point measured in Meters

is the front vertex power

Secondary Focal Point ()

is the secondary focal point measured in Meters

is the back vertex power

Equivalent Power

is the equivalent power

power of the front of the lens

power of the back of the lens

is the thickness measured in Meters

is the refractive index

Equivalent Focal Point()

is equivalent focal point measured in Meters

is the equivalent power

Image Distance

Linear Magnification

Image Height

Now we are armed with the information we need to trace a thick lens.

Example: A plastic( n = 1.50) lens with a thickness of 20mm is surrounded by air. It has a front power of +20.00D and a back power of -3.00D. If and object is placed 100mm in front of the lens where does the image form, how much magnification is present, and is the image inverted or erect. Find the principal planes, equivalent power, linear magnification, image distance, and image height.

Step 1: Find the radius of curvature for both surfaces.

* Note the refracting ray is in air and the incident ray is in the material

Step 2: Find the back and front vertex powers.

Step 3: Find the equivalent power.

Step 4: Find the primary, secondary, and equivalent focal points.

Now we are ready to draw the lens, optical axis, principal planes, , and . First you need to draw the optical axis, grab a compass and draw (25mm).

Second from the apex of the 1^st surface measure and draw a point that is equal to the thickness of your lens(20mm) and with your compass draw (167mm) from that point.

Now that we have the lens drawn we need to mark our primary focal point, secondary focal point, and our principal planes on our optical axis.

We are now ready to finish drawing our optical axis, but we need to know how long to draw it. We know that our object is located -100mm from the front vertex, however we will measure it from p1.

Take the difference between f1 to front vertex and p1 to f1. Then subtract that number from your object distance.

-58 - (-56) = -2

-100 - (-2) = -98

-98 is the distance from p1 to your object or your object distance. Now we will use our equivalent power to find the image distance

So the image is located131mm from p2. Now let us find linear magnification.

So now we will trace three rays to show that our image ends up 131mm from p1, it will be inverted, and it will be magnified. We know this because of the result of our linear magnification formula.